WEBVTT 00:00:00.000 --> 00:00:00.988 align:middle line:90% [SQUEAKING] 00:00:00.988 --> 00:00:02.470 align:middle line:90% [RUSTLING] 00:00:02.470 --> 00:00:04.940 align:middle line:90% [CLICKING] 00:00:04.940 --> 00:00:14.870 align:middle line:90% 00:00:14.870 --> 00:00:17.090 align:middle line:84% BRYNMOR CHAPMAN: Can people hear me in the back? 00:00:17.090 --> 00:00:17.820 align:middle line:90% We good? 00:00:17.820 --> 00:00:18.440 align:middle line:90% OK. 00:00:18.440 --> 00:00:19.020 align:middle line:90% Nice. 00:00:19.020 --> 00:00:20.720 align:middle line:90% Let's get started. 00:00:20.720 --> 00:00:24.060 align:middle line:90% So last week, last week? 00:00:24.060 --> 00:00:28.880 align:middle line:84% Yes, last week, we started talking about our number theory 00:00:28.880 --> 00:00:31.460 align:middle line:84% unit and in particular, divisibility. 00:00:31.460 --> 00:00:34.970 align:middle line:84% And we saw a very exciting application of divisibility, 00:00:34.970 --> 00:00:36.560 align:middle line:90% saving New York City. 00:00:36.560 --> 00:00:39.230 align:middle line:84% Today we're going to look at an even more exciting example, 00:00:39.230 --> 00:00:41.540 align:middle line:90% modular arithmetic. 00:00:41.540 --> 00:00:50.730 align:middle line:84% So recall, so we saw Bézout's identity and the pulverizer. 00:00:50.730 --> 00:00:53.460 align:middle line:90% So I'll just write that up here. 00:00:53.460 --> 00:01:10.890 align:middle line:84% So for all integers a and b, there exists s and t such that-- 00:01:10.890 --> 00:01:13.680 align:middle line:84% that's a little bit confusing, sorry-- 00:01:13.680 --> 00:01:22.200 align:middle line:84% gcd of a, comma b equals as plus bt. 00:01:22.200 --> 00:01:26.400 align:middle line:84% And moreover, we can compute these efficiently, 00:01:26.400 --> 00:01:31.608 align:middle line:84% using the pulverizer, the extended Euclidean algorithm. 00:01:31.608 --> 00:01:33.150 align:middle line:84% Then the other useful fact that we're 00:01:33.150 --> 00:01:39.870 align:middle line:84% going to make use of today, every number 00:01:39.870 --> 00:01:55.380 align:middle line:84% can be written as an ILC of a and b if and only 00:01:55.380 --> 00:01:58.905 align:middle line:90% if it is divisible by their gcd. 00:01:58.905 --> 00:02:11.860 align:middle line:90% 00:02:11.860 --> 00:02:17.360 align:middle line:84% So we are going to make use of these two facts. 00:02:17.360 --> 00:02:18.210 align:middle line:90% So do not forget. 00:02:18.210 --> 00:02:21.010 align:middle line:90% 00:02:21.010 --> 00:02:23.470 align:middle line:90% Let's see. 00:02:23.470 --> 00:02:28.840 align:middle line:90% Oh, sorry about that. 00:02:28.840 --> 00:02:33.504 align:middle line:90% 00:02:33.504 --> 00:02:37.150 align:middle line:84% So let's start talking about modular arithmetic then. 00:02:37.150 --> 00:02:39.920 align:middle line:84% I expect that at least in some form, 00:02:39.920 --> 00:02:43.780 align:middle line:84% it is something that's fairly familiar to all of you. 00:02:43.780 --> 00:02:46.130 align:middle line:84% So presumably you learned back in grade school, 00:02:46.130 --> 00:02:49.750 align:middle line:84% what is the sum of an even integer and an odd integer? 00:02:49.750 --> 00:02:50.865 align:middle line:90% What do you know about it? 00:02:50.865 --> 00:02:54.550 align:middle line:90% 00:02:54.550 --> 00:02:56.150 align:middle line:90% Did people not learn this? 00:02:56.150 --> 00:02:57.020 align:middle line:90% Yeah? 00:02:57.020 --> 00:02:57.812 align:middle line:90% AUDIENCE: It's odd. 00:02:57.812 --> 00:02:59.152 align:middle line:90% BRYNMOR CHAPMAN: Yeah, it's odd. 00:02:59.152 --> 00:03:01.675 align:middle line:84% So, first example, even plus odd is odd. 00:03:01.675 --> 00:03:04.730 align:middle line:90% 00:03:04.730 --> 00:03:08.380 align:middle line:90% Even plus odd is odd. 00:03:08.380 --> 00:03:12.080 align:middle line:90% 00:03:12.080 --> 00:03:19.413 align:middle line:84% What is the last digit of, say, 357 by 994? 00:03:19.413 --> 00:03:22.880 align:middle line:90% 00:03:22.880 --> 00:03:23.510 align:middle line:90% Yeah? 00:03:23.510 --> 00:03:24.410 align:middle line:90% AUDIENCE: 8. 00:03:24.410 --> 00:03:27.685 align:middle line:84% BRYNMOR CHAPMAN: 8, yeah, so ends in 8. 00:03:27.685 --> 00:03:31.200 align:middle line:90% 00:03:31.200 --> 00:03:32.000 align:middle line:90% Why is that? 00:03:32.000 --> 00:03:34.593 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:03:34.593 --> 00:03:37.010 align:middle line:84% BRYNMOR CHAPMAN: Yeah, you can basically ignore everything 00:03:37.010 --> 00:03:39.410 align:middle line:90% except the last two digits. 00:03:39.410 --> 00:03:40.620 align:middle line:90% You multiply those together. 00:03:40.620 --> 00:03:41.280 align:middle line:90% You get 28. 00:03:41.280 --> 00:03:42.210 align:middle line:90% That ends in an 8. 00:03:42.210 --> 00:03:45.386 align:middle line:84% So this is also going to end in an 8. 00:03:45.386 --> 00:03:47.940 align:middle line:90% It's currently 2:39 PM. 00:03:47.940 --> 00:03:49.710 align:middle line:84% What time will it be in 49 hours? 00:03:49.710 --> 00:03:52.806 align:middle line:90% 00:03:52.806 --> 00:03:54.180 align:middle line:90% AUDIENCE: 3:30. 00:03:54.180 --> 00:04:03.060 align:middle line:84% BRYNMOR CHAPMAN: 3:30 yeah, about, 3:39 and 49 hours. 00:04:03.060 --> 00:04:03.890 align:middle line:90% And why is that? 00:04:03.890 --> 00:04:07.140 align:middle line:90% 00:04:07.140 --> 00:04:10.260 align:middle line:90% Yeah, someone else say? 00:04:10.260 --> 00:04:11.530 align:middle line:90% How are we getting to that? 00:04:11.530 --> 00:04:12.804 align:middle line:90% Yeah? 00:04:12.804 --> 00:04:13.793 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:04:13.793 --> 00:04:14.710 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:04:14.710 --> 00:04:17.920 align:middle line:84% So 49 hours is one hour more than two full days. 00:04:17.920 --> 00:04:19.560 align:middle line:84% We can forget about those two days 00:04:19.560 --> 00:04:23.100 align:middle line:84% and just add an hour onto the current time. 00:04:23.100 --> 00:04:25.565 align:middle line:84% What day of the week will it be 100 days from now? 00:04:25.565 --> 00:04:28.702 align:middle line:90% 00:04:28.702 --> 00:04:29.640 align:middle line:90% AUDIENCE: Thursday. 00:04:29.640 --> 00:04:31.030 align:middle line:90% BRYNMOR CHAPMAN: Thursday, yeah. 00:04:31.030 --> 00:04:35.950 align:middle line:84% 100 days is two days more than 14 weeks. 00:04:35.950 --> 00:04:38.290 align:middle line:84% So we can forget about those 14 weeks. 00:04:38.290 --> 00:04:41.080 align:middle line:84% Add those two days, and we get Thursday. 00:04:41.080 --> 00:04:45.090 align:middle line:84% So these are all fairly familiar examples to most of you, 00:04:45.090 --> 00:04:48.520 align:middle line:84% hopefully, of modular arithmetic. 00:04:48.520 --> 00:04:53.420 align:middle line:84% And the theme is that when you're working modulo n, 00:04:53.420 --> 00:04:56.090 align:middle line:84% you're just ignoring all multiples of n 00:04:56.090 --> 00:04:59.460 align:middle line:90% and only focusing on remainders. 00:04:59.460 --> 00:05:03.350 align:middle line:84% So even and odd, this is the remainder when dividing by 2. 00:05:03.350 --> 00:05:05.340 align:middle line:84% Days of the week, we divide by 7. 00:05:05.340 --> 00:05:07.950 align:middle line:84% Look at the remainder, last digit. 00:05:07.950 --> 00:05:11.420 align:middle line:84% This is remainder when we divide by 10; 00:05:11.420 --> 00:05:14.150 align:middle line:84% hour remainder when we divide by 12 or 24, 00:05:14.150 --> 00:05:17.180 align:middle line:84% depending on whether you care about AM or PM. 00:05:17.180 --> 00:05:20.420 align:middle line:84% And so we often call modular arithmetic clock arithmetic 00:05:20.420 --> 00:05:21.673 align:middle line:90% for this reason. 00:05:21.673 --> 00:05:23.090 align:middle line:84% Because it's something that people 00:05:23.090 --> 00:05:25.230 align:middle line:84% are familiar with in that context. 00:05:25.230 --> 00:05:29.180 align:middle line:90% So let's define what we mean. 00:05:29.180 --> 00:05:31.180 align:middle line:84% Actually, why don't we use a new board for this. 00:05:31.180 --> 00:05:36.440 align:middle line:90% 00:05:36.440 --> 00:05:40.510 align:middle line:84% So we're going to say that a is congruent. 00:05:40.510 --> 00:05:42.740 align:middle line:90% We want n to b. 00:05:42.740 --> 00:05:46.760 align:middle line:84% So this will usually be pronounced 00:05:46.760 --> 00:05:56.695 align:middle line:84% a is congruent to b modulo n if and only if n divides a minus b. 00:05:56.695 --> 00:06:02.642 align:middle line:90% 00:06:02.642 --> 00:06:08.060 align:middle line:84% So you might also see this written as a congruent 00:06:08.060 --> 00:06:11.690 align:middle line:90% to b mod n. 00:06:11.690 --> 00:06:14.670 align:middle line:84% Who has seen this notation before? 00:06:14.670 --> 00:06:15.170 align:middle line:90% Many. 00:06:15.170 --> 00:06:18.290 align:middle line:90% Who has seen this notation? 00:06:18.290 --> 00:06:20.812 align:middle line:90% A couple, not quite as many. 00:06:20.812 --> 00:06:23.490 align:middle line:84% So for the purposes of this class, 00:06:23.490 --> 00:06:26.130 align:middle line:84% we're going to generally be using this notation. 00:06:26.130 --> 00:06:30.186 align:middle line:84% We're going to try not to use the right one. 00:06:30.186 --> 00:06:32.705 align:middle line:90% Yes, right. 00:06:32.705 --> 00:06:36.440 align:middle line:84% And I will explain why in a few minutes. 00:06:36.440 --> 00:06:38.330 align:middle line:90% The right one, Even. 00:06:38.330 --> 00:06:40.220 align:middle line:84% For people who think they're familiar, 00:06:40.220 --> 00:06:44.932 align:middle line:84% it often leads to some confusion. 00:06:44.932 --> 00:06:46.730 align:middle line:84% So let's try and stick with this one 00:06:46.730 --> 00:06:49.074 align:middle line:84% until you really know what you're doing. 00:06:49.074 --> 00:06:52.130 align:middle line:84% But for future reference, if you see the right one, 00:06:52.130 --> 00:06:53.130 align:middle line:90% it means the same thing. 00:06:53.130 --> 00:06:56.284 align:middle line:90% 00:06:56.284 --> 00:07:01.870 align:middle line:84% So basically, we would like to consider a and b to be the same. 00:07:01.870 --> 00:07:03.910 align:middle line:84% You can think of this, roughly speaking, 00:07:03.910 --> 00:07:09.580 align:middle line:84% as equals if n divides their difference. 00:07:09.580 --> 00:07:13.300 align:middle line:84% So for example, if we're working modulo 5, 00:07:13.300 --> 00:07:18.860 align:middle line:84% there are only five different distinct values. 00:07:18.860 --> 00:07:25.420 align:middle line:84% So we're going to notate them like this. 00:07:25.420 --> 00:07:32.260 align:middle line:84% So this is the congruence class of 0 modulo 5, so everything 00:07:32.260 --> 00:07:34.160 align:middle line:90% that's congruent to 0. 00:07:34.160 --> 00:07:39.790 align:middle line:84% So this is going to be 0 plus or minus 5 plus or minus 00:07:39.790 --> 00:07:42.760 align:middle line:90% 10, et cetera. 00:07:42.760 --> 00:07:47.160 align:middle line:84% So that's one set of values that are the same. 00:07:47.160 --> 00:07:49.700 align:middle line:90% 00:07:49.700 --> 00:07:51.860 align:middle line:84% Got a second one, represented by 1. 00:07:51.860 --> 00:07:58.550 align:middle line:84% And this is going to be 1, 6, 11, minus 4, 00:07:58.550 --> 00:08:08.720 align:middle line:84% minus 9, so everything that's equivalent to 1 modulo 5. 00:08:08.720 --> 00:08:17.720 align:middle line:84% And then we've got three more defined analogously. 00:08:17.720 --> 00:08:21.800 align:middle line:84% And so for a general k, which of these five groups 00:08:21.800 --> 00:08:23.870 align:middle line:90% is it going to belong to? 00:08:23.870 --> 00:08:25.060 align:middle line:90% How would we figure it out? 00:08:25.060 --> 00:08:27.680 align:middle line:90% 00:08:27.680 --> 00:08:29.090 align:middle line:90% Yeah? 00:08:29.090 --> 00:08:31.790 align:middle line:84% AUDIENCE: Divided by [INAUDIBLE]. 00:08:31.790 --> 00:08:34.100 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so the answer was we're 00:08:34.100 --> 00:08:36.200 align:middle line:84% going to divide it by the modulus 5 00:08:36.200 --> 00:08:39.059 align:middle line:90% and look at the remainder. 00:08:39.059 --> 00:08:48.820 align:middle line:84% So a general k is going to be an element of the remainder 00:08:48.820 --> 00:08:50.415 align:middle line:90% when we divide k by 5. 00:08:50.415 --> 00:08:53.635 align:middle line:90% 00:08:53.635 --> 00:08:57.810 align:middle line:84% Does that make sense to everybody? 00:08:57.810 --> 00:09:05.620 align:middle line:84% So in particular, if we write k equals 5q plus r, 00:09:05.620 --> 00:09:13.300 align:middle line:84% where r is between 0 and 5, then k is going to be an element 00:09:13.300 --> 00:09:18.376 align:middle line:90% of this set. 00:09:18.376 --> 00:09:21.640 align:middle line:84% Does that makes sense to everybody? 00:09:21.640 --> 00:09:24.280 align:middle line:90% Reasonably happy with this? 00:09:24.280 --> 00:09:24.780 align:middle line:90% OK. 00:09:24.780 --> 00:09:31.932 align:middle line:90% 00:09:31.932 --> 00:09:34.275 align:middle line:84% So more generally, recall the division theorem. 00:09:34.275 --> 00:09:42.700 align:middle line:90% 00:09:42.700 --> 00:09:48.400 align:middle line:84% So the division theorem says, for all pairs n, 00:09:48.400 --> 00:09:57.490 align:middle line:84% d with d greater than 0, there exists 00:09:57.490 --> 00:10:04.720 align:middle line:84% a unique pair of integers q and r with 0 less than 00:10:04.720 --> 00:10:18.717 align:middle line:84% or equal to r less than d such that n equals qd plus r. 00:10:18.717 --> 00:10:21.250 align:middle line:90% 00:10:21.250 --> 00:10:23.973 align:middle line:84% And we say that q here is the quotient. 00:10:23.973 --> 00:10:30.310 align:middle line:90% 00:10:30.310 --> 00:10:36.730 align:middle line:84% And we sometimes denote this as div n, d. 00:10:36.730 --> 00:10:44.090 align:middle line:84% Or sometimes, we might write this in infix notation, n div d. 00:10:44.090 --> 00:10:46.670 align:middle line:90% And r is the remainder. 00:10:46.670 --> 00:10:52.730 align:middle line:90% So this is going to be rem n, d. 00:10:52.730 --> 00:10:59.460 align:middle line:84% Or sometimes, we also write this as n rem d. 00:10:59.460 --> 00:11:00.368 align:middle line:90% Yeah? 00:11:00.368 --> 00:11:01.527 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:11:01.527 --> 00:11:02.485 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 00:11:02.485 --> 00:11:04.318 align:middle line:84% AUDIENCE: The exclamation after [INAUDIBLE]. 00:11:04.318 --> 00:11:06.930 align:middle line:84% BRYNMOR CHAPMAN: Oh, the exclamation after the exists. 00:11:06.930 --> 00:11:10.280 align:middle line:84% This means there exists a unique pair. 00:11:10.280 --> 00:11:14.120 align:middle line:84% So not only does there exists some pair, but there's only one 00:11:14.120 --> 00:11:14.620 align:middle line:90% of them. 00:11:14.620 --> 00:11:20.800 align:middle line:90% 00:11:20.800 --> 00:11:23.900 align:middle line:84% So when working modulo n, a number 00:11:23.900 --> 00:11:26.520 align:middle line:84% is always going to be equivalent to its remainder. 00:11:26.520 --> 00:11:30.560 align:middle line:84% Sometimes we'll also call this its residue. 00:11:30.560 --> 00:11:37.940 align:middle line:84% And just as we saw up here, these residue classes 00:11:37.940 --> 00:11:43.060 align:middle line:84% form a partition of all of the integers. 00:11:43.060 --> 00:11:44.640 align:middle line:90% Oh, sorry. 00:11:44.640 --> 00:11:49.140 align:middle line:84% So the number is 0 up through n minus 1. 00:11:49.140 --> 00:11:54.255 align:middle line:84% These represent all of the possible groupings modulo n. 00:11:54.255 --> 00:12:00.270 align:middle line:90% 00:12:00.270 --> 00:12:01.960 align:middle line:84% Does that make sense to everybody? 00:12:01.960 --> 00:12:19.530 align:middle line:90% 00:12:19.530 --> 00:12:23.610 align:middle line:84% So the following theorem is quite useful. 00:12:23.610 --> 00:12:30.240 align:middle line:84% So a and b are congruent modulo n 00:12:30.240 --> 00:12:41.290 align:middle line:84% if and only if the remainder of a, n is equal to the remainder 00:12:41.290 --> 00:12:42.975 align:middle line:90% when you divide b by n. 00:12:42.975 --> 00:12:48.930 align:middle line:90% 00:12:48.930 --> 00:12:51.470 align:middle line:84% So how might we prove this rigorously? 00:12:51.470 --> 00:12:52.050 align:middle line:90% Any ideas? 00:12:52.050 --> 00:13:05.830 align:middle line:90% 00:13:05.830 --> 00:13:06.670 align:middle line:90% Somebody? 00:13:06.670 --> 00:13:07.340 align:middle line:90% Volunteers? 00:13:07.340 --> 00:13:09.910 align:middle line:90% Yeah? 00:13:09.910 --> 00:13:12.940 align:middle line:84% AUDIENCE: [INAUDIBLE] a is q bar. 00:13:12.940 --> 00:13:15.520 align:middle line:90% 00:13:15.520 --> 00:13:18.190 align:middle line:90% And do the same for each. 00:13:18.190 --> 00:13:20.130 align:middle line:90% You can have q [INAUDIBLE]. 00:13:20.130 --> 00:13:24.003 align:middle line:90% 00:13:24.003 --> 00:13:24.920 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:13:24.920 --> 00:13:27.442 align:middle line:84% So we're going to use the division theorem. 00:13:27.442 --> 00:13:30.740 align:middle line:84% If we apply the division theorem to both a and b, 00:13:30.740 --> 00:13:32.860 align:middle line:90% we're going to get the same r. 00:13:32.860 --> 00:13:36.310 align:middle line:84% I'm going to back up a step first. 00:13:36.310 --> 00:13:38.150 align:middle line:84% At a very high level, what do we need 00:13:38.150 --> 00:13:41.433 align:middle line:84% to do in order to prove a statement like this? 00:13:41.433 --> 00:13:43.100 align:middle line:84% We're trying to prove an if and only if. 00:13:43.100 --> 00:13:46.352 align:middle line:84% We have to prove implications in both directions. 00:13:46.352 --> 00:13:57.920 align:middle line:84% So first direction, if rem a, n equals rem b, n, 00:13:57.920 --> 00:14:03.514 align:middle line:90% then a is congruent mod n to b. 00:14:03.514 --> 00:14:06.260 align:middle line:90% 00:14:06.260 --> 00:14:11.250 align:middle line:84% So we're going to assume that these are equal. 00:14:11.250 --> 00:14:13.670 align:middle line:84% So then the r that we get from the division theorem 00:14:13.670 --> 00:14:15.380 align:middle line:90% is the same in both cases. 00:14:15.380 --> 00:14:20.510 align:middle line:90% So a equals qn plus r. 00:14:20.510 --> 00:14:23.741 align:middle line:90% b equals q prime n plus r. 00:14:23.741 --> 00:14:27.920 align:middle line:90% 00:14:27.920 --> 00:14:30.860 align:middle line:84% Now what happens when we subtract them? 00:14:30.860 --> 00:14:37.380 align:middle line:84% Well, then we get a minus b equals n times q 00:14:37.380 --> 00:14:41.800 align:middle line:90% minus q prime plus r minus r. 00:14:41.800 --> 00:14:42.645 align:middle line:90% And this is 0. 00:14:42.645 --> 00:14:45.150 align:middle line:90% 00:14:45.150 --> 00:14:49.630 align:middle line:84% So now the difference is a multiple of n. 00:14:49.630 --> 00:14:55.450 align:middle line:84% So by our definition, a and b are congruent. 00:14:55.450 --> 00:14:59.340 align:middle line:84% Does that make sense to everybody? 00:14:59.340 --> 00:15:00.060 align:middle line:90% Yeah? 00:15:00.060 --> 00:15:01.860 align:middle line:84% AUDIENCE: Can [INAUDIBLE] examples 00:15:01.860 --> 00:15:05.910 align:middle line:84% of a, and b in certain regular numbers? 00:15:05.910 --> 00:15:08.285 align:middle line:84% BRYNMOR CHAPMAN: Oh, so some examples of a, n and b? 00:15:08.285 --> 00:15:08.910 align:middle line:90% AUDIENCE: Yeah. 00:15:08.910 --> 00:15:10.410 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:15:10.410 --> 00:15:19.290 align:middle line:84% So, for example, 17 is congruent mod 5 to 12. 00:15:19.290 --> 00:15:21.730 align:middle line:84% Because 5 divides the difference. 00:15:21.730 --> 00:15:23.730 align:middle line:90% If you subtract them, you get 5. 00:15:23.730 --> 00:15:25.720 align:middle line:90% 5 is a multiple of 5. 00:15:25.720 --> 00:15:26.860 align:middle line:90% So they're congruent. 00:15:26.860 --> 00:15:30.000 align:middle line:90% 00:15:30.000 --> 00:15:35.950 align:middle line:84% 17 is also congruent to, say, 32. 00:15:35.950 --> 00:15:39.040 align:middle line:84% If you subtract these, you get negative 15-- same deal. 00:15:39.040 --> 00:15:44.040 align:middle line:90% It's not congruent to 33. 00:15:44.040 --> 00:15:47.360 align:middle line:90% 00:15:47.360 --> 00:15:49.420 align:middle line:84% If you subtract those, you get negative 16, 00:15:49.420 --> 00:15:51.405 align:middle line:90% which is not a multiple of 5. 00:15:51.405 --> 00:15:54.920 align:middle line:90% 00:15:54.920 --> 00:15:56.066 align:middle line:90% Any other questions? 00:15:56.066 --> 00:15:59.260 align:middle line:90% 00:15:59.260 --> 00:15:59.980 align:middle line:90% Yeah? 00:15:59.980 --> 00:16:01.339 align:middle line:90% AUDIENCE: [INAUDIBLE]? 00:16:01.339 --> 00:16:04.160 align:middle line:90% 00:16:04.160 --> 00:16:06.410 align:middle line:84% BRYNMOR CHAPMAN: So this is only half the proof. 00:16:06.410 --> 00:16:08.090 align:middle line:90% We're proving one direction. 00:16:08.090 --> 00:16:12.430 align:middle line:84% So we are assuming that the two remainders are the same. 00:16:12.430 --> 00:16:16.570 align:middle line:90% And we're trying to prove this. 00:16:16.570 --> 00:16:18.830 align:middle line:84% So if the two remainders are the same, 00:16:18.830 --> 00:16:20.780 align:middle line:84% that's from the division theorem. 00:16:20.780 --> 00:16:23.380 align:middle line:84% We've got the same r in both cases. 00:16:23.380 --> 00:16:27.590 align:middle line:84% So we're going to write both numbers in that form. 00:16:27.590 --> 00:16:30.706 align:middle line:90% So a is qn plus r. 00:16:30.706 --> 00:16:33.060 align:middle line:90% b Is q prime n plus r prime. 00:16:33.060 --> 00:16:35.850 align:middle line:84% But we established that r is equal to r prime. 00:16:35.850 --> 00:16:40.280 align:middle line:84% So we're just writing that as q prime n plus r. 00:16:40.280 --> 00:16:44.040 align:middle line:84% And now in order to show that a and b are equivalent, 00:16:44.040 --> 00:16:47.180 align:middle line:84% we have to subtract and see whether the difference is 00:16:47.180 --> 00:16:49.160 align:middle line:90% a multiple of n. 00:16:49.160 --> 00:16:54.860 align:middle line:84% So in this case, when we subtract them, the r's cancel. 00:16:54.860 --> 00:17:00.590 align:middle line:84% The n factors out, and we get n times q minus q prime. 00:17:00.590 --> 00:17:03.060 align:middle line:84% So that is indeed a multiple of n. 00:17:03.060 --> 00:17:05.910 align:middle line:90% And so n divides the difference. 00:17:05.910 --> 00:17:12.329 align:middle line:84% So they're equivalent, congruent. 00:17:12.329 --> 00:17:13.190 align:middle line:90% Yeah? 00:17:13.190 --> 00:17:15.315 align:middle line:84% AUDIENCE: Can you just please repeat how you worked 00:17:15.315 --> 00:17:18.920 align:middle line:84% [INAUDIBLE] that notation with congruencies? 00:17:18.920 --> 00:17:20.375 align:middle line:84% BRYNMOR CHAPMAN: How we say this? 00:17:20.375 --> 00:17:21.020 align:middle line:90% AUDIENCE: Yeah. 00:17:21.020 --> 00:17:23.000 align:middle line:84% BRYNMOR CHAPMAN: a is congruent to b mod n. 00:17:23.000 --> 00:17:25.430 align:middle line:90% Or a is congruent mod n to b. 00:17:25.430 --> 00:17:28.525 align:middle line:84% There are several different ways of saying it. 00:17:28.525 --> 00:17:41.260 align:middle line:90% 00:17:41.260 --> 00:17:42.940 align:middle line:84% What about going in the other direction? 00:17:42.940 --> 00:17:45.690 align:middle line:90% 00:17:45.690 --> 00:17:56.760 align:middle line:84% Suppose a is congruent mod n to the b. 00:17:56.760 --> 00:17:58.980 align:middle line:84% Now we want to prove that the remainders, when 00:17:58.980 --> 00:18:00.460 align:middle line:90% you divide by n, are the same. 00:18:00.460 --> 00:18:01.355 align:middle line:90% How do we do that? 00:18:01.355 --> 00:18:07.800 align:middle line:90% 00:18:07.800 --> 00:18:08.700 align:middle line:90% Yeah? 00:18:08.700 --> 00:18:13.950 align:middle line:84% AUDIENCE: We know that we might say for [INAUDIBLE], some k 00:18:13.950 --> 00:18:16.011 align:middle line:90% times n. 00:18:16.011 --> 00:18:19.290 align:middle line:84% Could we add that difference to one of them? 00:18:19.290 --> 00:18:29.336 align:middle line:84% Could we take b minus n [INAUDIBLE] and add that to b. 00:18:29.336 --> 00:18:32.480 align:middle line:84% Then we should get a, then n, I think-- 00:18:32.480 --> 00:18:35.110 align:middle line:90% 00:18:35.110 --> 00:18:37.903 align:middle line:84% and then we'll have a equals [INAUDIBLE]. 00:18:37.903 --> 00:18:46.293 align:middle line:90% 00:18:46.293 --> 00:18:47.210 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:18:47.210 --> 00:18:51.632 align:middle line:84% So the last step here is now to use the division theorem again. 00:18:51.632 --> 00:18:53.440 align:middle line:84% So if we apply the division theorem, 00:18:53.440 --> 00:18:56.150 align:middle line:84% I guess, to b is probably going to be the most useful. 00:18:56.150 --> 00:18:59.030 align:middle line:90% We can write b equals-- 00:18:59.030 --> 00:19:03.324 align:middle line:84% let's do the same form that we had there, q prime n plus r. 00:19:03.324 --> 00:19:06.610 align:middle line:90% 00:19:06.610 --> 00:19:10.810 align:middle line:90% Now, a is b plus kn. 00:19:10.810 --> 00:19:17.530 align:middle line:84% So that's going to be q prime plus k times n plus r. 00:19:17.530 --> 00:19:20.560 align:middle line:90% 00:19:20.560 --> 00:19:24.160 align:middle line:84% And now the division theorem says that this q prime plus 00:19:24.160 --> 00:19:26.630 align:middle line:90% k and r, those are unique. 00:19:26.630 --> 00:19:30.150 align:middle line:84% So that q prime plus k is the q that we need. 00:19:30.150 --> 00:19:34.712 align:middle line:90% And the r is the same. 00:19:34.712 --> 00:19:38.220 align:middle line:84% So both of them have the same remainder. 00:19:38.220 --> 00:19:41.420 align:middle line:90% 00:19:41.420 --> 00:19:43.620 align:middle line:84% Does this make sense to everybody? 00:19:43.620 --> 00:19:44.708 align:middle line:90% Yeah? 00:19:44.708 --> 00:19:49.310 align:middle line:84% AUDIENCE: I'm just wondering how you want to show that a is-- 00:19:49.310 --> 00:19:54.133 align:middle line:84% a is congruent to n, the n mod [INAUDIBLE]. 00:19:54.133 --> 00:19:55.800 align:middle line:84% BRYNMOR CHAPMAN: a congruent to b mod n? 00:19:55.800 --> 00:19:57.020 align:middle line:90% AUDIENCE: Yes. 00:19:57.020 --> 00:19:59.562 align:middle line:84% BRYNMOR CHAPMAN: So what we're trying to prove in if and only 00:19:59.562 --> 00:20:04.130 align:middle line:84% if here, so for one direction, we're assuming one side 00:20:04.130 --> 00:20:05.490 align:middle line:90% and trying to prove the other. 00:20:05.490 --> 00:20:09.180 align:middle line:84% And for the other direction, we do the opposite. 00:20:09.180 --> 00:20:11.150 align:middle line:90% So this is one direction. 00:20:11.150 --> 00:20:14.870 align:middle line:84% We're assuming that the remainders are the same. 00:20:14.870 --> 00:20:17.280 align:middle line:84% And we want to prove that they're equivalent, 00:20:17.280 --> 00:20:18.650 align:middle line:90% they're congruent. 00:20:18.650 --> 00:20:21.180 align:middle line:84% For the other direction, we assume that they're congruent. 00:20:21.180 --> 00:20:25.462 align:middle line:84% And then we want to prove that the remainders are the same, so 00:20:25.462 --> 00:20:26.295 align:middle line:90% two separate proofs. 00:20:26.295 --> 00:20:32.050 align:middle line:90% 00:20:32.050 --> 00:20:33.530 align:middle line:90% Any other questions? 00:20:33.530 --> 00:20:39.770 align:middle line:90% 00:20:39.770 --> 00:20:43.720 align:middle line:84% In some sense, we can consider the n different numbers, like 0 00:20:43.720 --> 00:20:47.080 align:middle line:84% through n minus 1, to be the only numbers we 00:20:47.080 --> 00:20:49.053 align:middle line:90% care about when working mod n. 00:20:49.053 --> 00:20:51.470 align:middle line:84% Everything else is going to be equivalent to one of those. 00:20:51.470 --> 00:20:53.345 align:middle line:84% It's going to be equivalent to its remainder. 00:20:53.345 --> 00:20:56.530 align:middle line:90% 00:20:56.530 --> 00:20:59.413 align:middle line:84% So I'm going to take a moment here to talk about notation. 00:20:59.413 --> 00:21:13.960 align:middle line:90% 00:21:13.960 --> 00:21:19.380 align:middle line:84% So the remainder, which we've called rem a, comma n-- 00:21:19.380 --> 00:21:27.320 align:middle line:90% 00:21:27.320 --> 00:21:31.350 align:middle line:90% so our convention. 00:21:31.350 --> 00:21:36.560 align:middle line:84% Is that even if n is negative, this remainder should always 00:21:36.560 --> 00:21:38.540 align:middle line:90% be positive. 00:21:38.540 --> 00:21:41.300 align:middle line:84% Not everybody uses this convention. 00:21:41.300 --> 00:21:43.610 align:middle line:84% So in particular, when you're dealing 00:21:43.610 --> 00:21:49.130 align:middle line:84% with programming languages, most have some operation 00:21:49.130 --> 00:21:52.580 align:middle line:90% that looks like this. 00:21:52.580 --> 00:21:56.840 align:middle line:84% It may be denoted differently in different languages. 00:21:56.840 --> 00:21:58.470 align:middle line:90% Sometimes, it's a percent sign. 00:21:58.470 --> 00:22:01.700 align:middle line:84% Sometimes, it's actually rem or mod, or something like that. 00:22:01.700 --> 00:22:07.380 align:middle line:84% And sometimes, it'll behave the same as how we defined it. 00:22:07.380 --> 00:22:10.560 align:middle line:84% Sometimes, it'll be allowed to be negative. 00:22:10.560 --> 00:22:12.380 align:middle line:84% So if n is negative, the remainder 00:22:12.380 --> 00:22:15.200 align:middle line:84% might be negative or something like that. 00:22:15.200 --> 00:22:21.290 align:middle line:84% And some languages have both, usually with different names, 00:22:21.290 --> 00:22:25.470 align:middle line:90% like mod versus double mod. 00:22:25.470 --> 00:22:29.800 align:middle line:84% In this class, whenever we write rem a, comma n, 00:22:29.800 --> 00:22:34.300 align:middle line:84% we always mean the positive or the non-negative remainder. 00:22:34.300 --> 00:22:41.295 align:middle line:84% So this is always going to be between 0 and n. 00:22:41.295 --> 00:22:45.870 align:middle line:90% 00:22:45.870 --> 00:22:50.970 align:middle line:84% And perhaps even more confusing, sometimes this 00:22:50.970 --> 00:22:57.940 align:middle line:84% would be written as mod of a, comma n; or even worse, 00:22:57.940 --> 00:23:04.060 align:middle line:90% an infix notation, a mod n. 00:23:04.060 --> 00:23:08.040 align:middle line:84% So this is why we don't like to use the word "mod" 00:23:08.040 --> 00:23:14.610 align:middle line:84% when talking about modular congruences because if you see 00:23:14.610 --> 00:23:22.215 align:middle line:84% something like, say, a equals b mod n, what does that mean? 00:23:22.215 --> 00:23:29.927 align:middle line:90% 00:23:29.927 --> 00:23:31.260 align:middle line:90% Who can tell me what that means? 00:23:31.260 --> 00:23:36.910 align:middle line:90% 00:23:36.910 --> 00:23:37.750 align:middle line:90% Yeah? 00:23:37.750 --> 00:23:39.315 align:middle line:90% Do you have a guess? 00:23:39.315 --> 00:23:42.310 align:middle line:90% 00:23:42.310 --> 00:23:47.855 align:middle line:84% AUDIENCE: Oh, but I was thinking it was a [INAUDIBLE]. 00:23:47.855 --> 00:23:49.230 align:middle line:84% BRYNMOR CHAPMAN: And why is that? 00:23:49.230 --> 00:23:51.567 align:middle line:90% 00:23:51.567 --> 00:23:52.150 align:middle line:90% AUDIENCE: Why? 00:23:52.150 --> 00:23:56.810 align:middle line:90% 00:23:56.810 --> 00:23:59.650 align:middle line:90% So that's just the definition. 00:23:59.650 --> 00:24:01.030 align:middle line:84% BRYNMOR CHAPMAN: Yeah, the answer 00:24:01.030 --> 00:24:03.850 align:middle line:90% was that's just the definition. 00:24:03.850 --> 00:24:05.680 align:middle line:84% So the important distinction here 00:24:05.680 --> 00:24:07.813 align:middle line:90% is between these two things. 00:24:07.813 --> 00:24:12.160 align:middle line:90% 00:24:12.160 --> 00:24:14.000 align:middle line:84% These are very different statements. 00:24:14.000 --> 00:24:20.910 align:middle line:84% The mod here is attached to this equivalence sign. 00:24:20.910 --> 00:24:28.230 align:middle line:84% And so this is talking about a relation between a and b. 00:24:28.230 --> 00:24:32.890 align:middle line:84% This is a function written in infix notation. 00:24:32.890 --> 00:24:38.400 align:middle line:84% So this is saying that a is actually equal to the one value 00:24:38.400 --> 00:24:42.090 align:middle line:84% that you get when you divide b by n and take the remainder. 00:24:42.090 --> 00:24:44.620 align:middle line:90% So these mean different things. 00:24:44.620 --> 00:24:47.760 align:middle line:84% It is going to be much less confusing if you 00:24:47.760 --> 00:24:56.820 align:middle line:84% use a equals b rem n, or in prefix notation rem b, 00:24:56.820 --> 00:25:03.180 align:middle line:84% comma n for the left one, and a congruent mod 00:25:03.180 --> 00:25:06.681 align:middle line:90% n to be for the right one. 00:25:06.681 --> 00:25:08.970 align:middle line:84% So it's hopefully, much more obvious 00:25:08.970 --> 00:25:11.400 align:middle line:84% what the distinction is between these two things 00:25:11.400 --> 00:25:13.000 align:middle line:90% than between these two things. 00:25:13.000 --> 00:25:15.150 align:middle line:84% So please use the ones on the bottom 00:25:15.150 --> 00:25:18.970 align:middle line:84% until you are very comfortable with modular arithmetic. 00:25:18.970 --> 00:25:23.590 align:middle line:90% 00:25:23.590 --> 00:25:28.460 align:middle line:84% So let's go back to the actual content now. 00:25:28.460 --> 00:25:36.840 align:middle line:90% 00:25:36.840 --> 00:25:40.020 align:middle line:84% So as I promised before, we're actually 00:25:40.020 --> 00:25:41.800 align:middle line:84% going to be talking about arithmetic. 00:25:41.800 --> 00:25:45.441 align:middle line:90% So let's start on that. 00:25:45.441 --> 00:25:48.000 align:middle line:84% So the simple statement that we wrote up there 00:25:48.000 --> 00:25:53.040 align:middle line:84% on the left, even plus odd equals odd, 00:25:53.040 --> 00:25:57.270 align:middle line:84% in some sense, that's saying something profound. 00:25:57.270 --> 00:26:00.960 align:middle line:84% No matter which even number and which odd number 00:26:00.960 --> 00:26:03.070 align:middle line:84% you pick, if you add them together, 00:26:03.070 --> 00:26:05.100 align:middle line:84% you're always going to get an odd number. 00:26:05.100 --> 00:26:13.770 align:middle line:84% And, more generally, if we take any a congruent to, 00:26:13.770 --> 00:26:23.230 align:middle line:84% let's say, 3 mod 5 and any b congruent mod 5 to, say, 4, 00:26:23.230 --> 00:26:28.610 align:middle line:84% if you add them, what do you know about it? 00:26:28.610 --> 00:26:29.985 align:middle line:84% What do you know about their sum? 00:26:29.985 --> 00:26:39.280 align:middle line:90% 00:26:39.280 --> 00:26:39.850 align:middle line:90% Yeah? 00:26:39.850 --> 00:26:40.702 align:middle line:90% AUDIENCE: 2 mod 5. 00:26:40.702 --> 00:26:41.702 align:middle line:90% BRYNMOR CHAPMAN: Mm-hmm. 00:26:41.702 --> 00:26:44.950 align:middle line:90% 00:26:44.950 --> 00:26:49.670 align:middle line:84% So their sum is always going to be congruent to 2 modulo 5. 00:26:49.670 --> 00:26:56.783 align:middle line:84% Or perhaps an easier thing to see 00:26:56.783 --> 00:26:58.575 align:middle line:84% is it's always going to be equivalent to 7. 00:26:58.575 --> 00:27:01.630 align:middle line:90% 00:27:01.630 --> 00:27:03.855 align:middle line:84% Which is, in turn, congruent to 2. 00:27:03.855 --> 00:27:06.640 align:middle line:90% 00:27:06.640 --> 00:27:12.460 align:middle line:84% So adding a and b modulo 5, you get the same result 00:27:12.460 --> 00:27:16.216 align:middle line:84% as if you add these two equivalent remainders. 00:27:16.216 --> 00:27:23.256 align:middle line:90% 00:27:23.256 --> 00:27:24.640 align:middle line:90% So we have a theorem. 00:27:24.640 --> 00:27:27.320 align:middle line:90% 00:27:27.320 --> 00:27:37.040 align:middle line:84% Suppose that a and b are equivalent mod n. 00:27:37.040 --> 00:27:40.680 align:middle line:84% Then for all c, we have the following. 00:27:40.680 --> 00:27:43.640 align:middle line:90% 00:27:43.640 --> 00:27:44.880 align:middle line:90% I have space here. 00:27:44.880 --> 00:27:53.000 align:middle line:84% So a plus c is congruent to b plus c. 00:27:53.000 --> 00:28:00.110 align:middle line:84% a minus c is congruent to b minus c. 00:28:00.110 --> 00:28:05.840 align:middle line:84% c minus a is congruent to c minus b. 00:28:05.840 --> 00:28:12.055 align:middle line:84% And a times c is congruent to b times c. 00:28:12.055 --> 00:28:17.976 align:middle line:90% 00:28:17.976 --> 00:28:18.660 align:middle line:90% So As. 00:28:18.660 --> 00:28:24.240 align:middle line:84% You might expect, if you want to add, subtract, or multiply 00:28:24.240 --> 00:28:28.980 align:middle line:84% modulo n, you're allowed to replace things 00:28:28.980 --> 00:28:32.615 align:middle line:84% wherever you want with something it's equivalent to. 00:28:32.615 --> 00:28:35.388 align:middle line:90% 00:28:35.388 --> 00:28:40.350 align:middle line:84% So at any point during this computation, you can replace-- 00:28:40.350 --> 00:28:45.120 align:middle line:90% oh, sorry, different a and b. 00:28:45.120 --> 00:28:47.080 align:middle line:84% This is not relevant for the theorem statement. 00:28:47.080 --> 00:28:49.080 align:middle line:84% But if a and b are equivalent, then 00:28:49.080 --> 00:28:52.170 align:middle line:84% you can replace a's with b's wherever you want, 00:28:52.170 --> 00:28:57.125 align:middle line:84% and you'll get the same result. Does that make sense? 00:28:57.125 --> 00:29:08.550 align:middle line:90% 00:29:08.550 --> 00:29:19.185 align:middle line:84% So, question, what about a to the c and b to the c? 00:29:19.185 --> 00:29:31.900 align:middle line:90% 00:29:31.900 --> 00:29:34.680 align:middle line:84% So let's assume that c is positive for now. 00:29:34.680 --> 00:29:43.000 align:middle line:90% 00:29:43.000 --> 00:29:47.650 align:middle line:84% Can I say anything about exponentiation modulo n? 00:29:47.650 --> 00:29:51.910 align:middle line:84% If we assume that a and b are congruent modulo n, 00:29:51.910 --> 00:29:56.170 align:middle line:84% can we say anything about a to the c versus b to the c? 00:29:56.170 --> 00:29:58.000 align:middle line:90% Yeah? 00:29:58.000 --> 00:30:00.760 align:middle line:84% AUDIENCE: Would they be the same because the maximal value 00:30:00.760 --> 00:30:05.530 align:middle line:84% of a [INAUDIBLE] possible part from the [INAUDIBLE] 00:30:05.530 --> 00:30:07.973 align:middle line:90% part of that is the same? 00:30:07.973 --> 00:30:08.890 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:30:08.890 --> 00:30:12.800 align:middle line:84% So let me rephrase that slightly. 00:30:12.800 --> 00:30:17.520 align:middle line:84% You can prove it more formally by induction. 00:30:17.520 --> 00:30:20.340 align:middle line:90% So you can repeatedly multiply. 00:30:20.340 --> 00:30:24.440 align:middle line:84% Every multiplication is going to give you congruent values. 00:30:24.440 --> 00:30:27.260 align:middle line:84% So by the time you get to the power of c, 00:30:27.260 --> 00:30:30.740 align:middle line:84% they're still going to be congruent. 00:30:30.740 --> 00:30:43.190 align:middle line:84% So, yes, this is true and the proof, by induction on c. 00:30:43.190 --> 00:30:49.580 align:middle line:84% So our base case is going to be when c equals 1. 00:30:49.580 --> 00:30:56.155 align:middle line:84% In this case, our assumption is that a is congruent to b. 00:30:56.155 --> 00:30:58.410 align:middle line:90% So a to the 1 is congruent to a. 00:30:58.410 --> 00:31:00.840 align:middle line:84% Well, it's equal to a. b to the 1 is equal to b. 00:31:00.840 --> 00:31:03.980 align:middle line:90% So this is fine. 00:31:03.980 --> 00:31:11.100 align:middle line:84% Our inductive step, so assume that a to the c minus 1 00:31:11.100 --> 00:31:15.000 align:middle line:84% is congruent mod n to b to the c minus 1. 00:31:15.000 --> 00:31:19.715 align:middle line:84% Now, what happens when we increment our exponent? 00:31:19.715 --> 00:31:32.820 align:middle line:90% 00:31:32.820 --> 00:31:34.220 align:middle line:90% Well, what is a to the c? 00:31:34.220 --> 00:31:37.980 align:middle line:90% 00:31:37.980 --> 00:31:39.270 align:middle line:90% Yeah? 00:31:39.270 --> 00:31:40.450 align:middle line:90% AUDIENCE: a is [INAUDIBLE]. 00:31:40.450 --> 00:31:48.570 align:middle line:90% 00:31:48.570 --> 00:31:52.835 align:middle line:84% BRYNMOR CHAPMAN: So we can strip off a factor of a. 00:31:52.835 --> 00:31:57.450 align:middle line:90% 00:31:57.450 --> 00:32:01.196 align:middle line:84% Now we can replace each of these factors individually. 00:32:01.196 --> 00:32:04.242 align:middle line:90% 00:32:04.242 --> 00:32:07.480 align:middle line:84% So by our theorem up there, this is 00:32:07.480 --> 00:32:11.110 align:middle line:84% congruent to b to the c minus 1 times a. 00:32:11.110 --> 00:32:12.900 align:middle line:90% We can replace this factor. 00:32:12.900 --> 00:32:17.680 align:middle line:90% 00:32:17.680 --> 00:32:21.537 align:middle line:84% And now by our theorem assumption, 00:32:21.537 --> 00:32:22.995 align:middle line:84% we can replace this factor as well. 00:32:22.995 --> 00:32:31.660 align:middle line:90% 00:32:31.660 --> 00:32:34.400 align:middle line:84% Does that make sense to everybody? 00:32:34.400 --> 00:32:42.760 align:middle line:90% 00:32:42.760 --> 00:32:50.280 align:middle line:84% What about c to the a and c to the b? 00:32:50.280 --> 00:32:59.770 align:middle line:90% 00:32:59.770 --> 00:33:02.080 align:middle line:84% So we just saw that when we're working modulo n, 00:33:02.080 --> 00:33:06.000 align:middle line:84% we're allowed to replace the base of exponents as we like. 00:33:06.000 --> 00:33:09.120 align:middle line:84% Can we do the same thing with the exponents themselves? 00:33:09.120 --> 00:33:15.710 align:middle line:90% 00:33:15.710 --> 00:33:16.825 align:middle line:90% Show of hands for yes? 00:33:16.825 --> 00:33:20.090 align:middle line:90% 00:33:20.090 --> 00:33:22.882 align:middle line:90% Show of hands for no? 00:33:22.882 --> 00:33:24.840 align:middle line:84% No seems like it's winning out by popular vote, 00:33:24.840 --> 00:33:25.740 align:middle line:90% but not by much. 00:33:25.740 --> 00:33:28.310 align:middle line:90% There are a lot of abstentions. 00:33:28.310 --> 00:33:30.145 align:middle line:84% Can anybody give me a reason why or why not? 00:33:30.145 --> 00:33:34.470 align:middle line:90% 00:33:34.470 --> 00:33:34.970 align:middle line:90% Yeah? 00:33:34.970 --> 00:33:37.678 align:middle line:90% 00:33:37.678 --> 00:33:39.470 align:middle line:84% AUDIENCE: My reason is just because there's 00:33:39.470 --> 00:33:46.700 align:middle line:84% one example of 2 squared is not congruent mod 5 [INAUDIBLE]. 00:33:46.700 --> 00:33:50.840 align:middle line:84% BRYNMOR CHAPMAN: 2 squared is not congruent mod 5 to what? 00:33:50.840 --> 00:33:51.590 align:middle line:90% AUDIENCE: 2 cubed. 00:33:51.590 --> 00:33:52.632 align:middle line:90% BRYNMOR CHAPMAN: 2 cubed. 00:33:52.632 --> 00:33:56.146 align:middle line:90% 00:33:56.146 --> 00:33:58.700 align:middle line:90% OK, that is a true statement. 00:33:58.700 --> 00:33:59.315 align:middle line:90% Does that-- 00:33:59.315 --> 00:34:02.020 align:middle line:90% 00:34:02.020 --> 00:34:03.520 align:middle line:84% AUDIENCE: In this case, it would be, 00:34:03.520 --> 00:34:05.118 align:middle line:84% if you want to prove [INAUDIBLE] it'd 00:34:05.118 --> 00:34:07.290 align:middle line:90% have to be proof by induction. 00:34:07.290 --> 00:34:08.540 align:middle line:90% That's the thing that's false. 00:34:08.540 --> 00:34:09.923 align:middle line:84% You just need one counterexample. 00:34:09.923 --> 00:34:10.840 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:34:10.840 --> 00:34:14.199 align:middle line:84% So you are correct that we only need one counterexample. 00:34:14.199 --> 00:34:15.949 align:middle line:90% Is this a counterexample? 00:34:15.949 --> 00:34:24.520 align:middle line:90% 00:34:24.520 --> 00:34:26.570 align:middle line:90% So let's see. 00:34:26.570 --> 00:34:32.920 align:middle line:84% So the claim is that if a is congruent to b mod n, then c 00:34:32.920 --> 00:34:35.659 align:middle line:84% to the a is congruent to c to the b. 00:34:35.659 --> 00:34:37.630 align:middle line:84% We haven't satisfied our precondition here. 00:34:37.630 --> 00:34:41.389 align:middle line:90% 2 is not congruent to 3 mod 5. 00:34:41.389 --> 00:34:44.870 align:middle line:90% So you have the right idea. 00:34:44.870 --> 00:34:47.020 align:middle line:84% We do want to find a single counterexample that 00:34:47.020 --> 00:34:49.093 align:middle line:90% will suffice to disprove this. 00:34:49.093 --> 00:34:51.010 align:middle line:84% We're going to need a different counterexample 00:34:51.010 --> 00:34:53.253 align:middle line:90% if we want this to work. 00:34:53.253 --> 00:34:55.420 align:middle line:84% Does anybody else have an idea for a counterexample? 00:34:55.420 --> 00:34:58.815 align:middle line:84% Or failing that, does anybody have a proof? 00:34:58.815 --> 00:35:05.870 align:middle line:90% 00:35:05.870 --> 00:35:08.630 align:middle line:90% Yeah? 00:35:08.630 --> 00:35:13.400 align:middle line:84% AUDIENCE: I could if it's a's, but you 00:35:13.400 --> 00:35:17.630 align:middle line:84% could split up c to the b's in smaller components 00:35:17.630 --> 00:35:20.990 align:middle line:84% and then show that it's [INAUDIBLE]. 00:35:20.990 --> 00:35:22.160 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:35:22.160 --> 00:35:25.470 align:middle line:84% What are the smaller components, factors of c? 00:35:25.470 --> 00:35:26.930 align:middle line:90% AUDIENCE: Yeah, basically. 00:35:26.930 --> 00:35:29.120 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:35:29.120 --> 00:35:34.310 align:middle line:84% So what do you want to do with those factors of c? 00:35:34.310 --> 00:35:39.180 align:middle line:84% I suppose we could strip off, say, a is smaller than b. 00:35:39.180 --> 00:35:41.330 align:middle line:90% We could strip off a of them. 00:35:41.330 --> 00:35:43.310 align:middle line:84% AUDIENCE: And then whatever's left over, 00:35:43.310 --> 00:35:48.020 align:middle line:90% if that's mod whatever that is. 00:35:48.020 --> 00:35:49.205 align:middle line:90% That's true. 00:35:49.205 --> 00:35:52.490 align:middle line:84% And then we know [INAUDIBLE] the whole truth is true. 00:35:52.490 --> 00:35:53.795 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:35:53.795 --> 00:35:57.620 align:middle line:84% So you want to divide out a bunch of factors of c 00:35:57.620 --> 00:36:00.036 align:middle line:84% until we get something like this? 00:36:00.036 --> 00:36:02.790 align:middle line:90% 00:36:02.790 --> 00:36:03.382 align:middle line:90% AUDIENCE: Yes. 00:36:03.382 --> 00:36:04.215 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:36:04.215 --> 00:36:10.230 align:middle line:90% 00:36:10.230 --> 00:36:11.690 align:middle line:90% How do you divide mod n? 00:36:11.690 --> 00:36:29.950 align:middle line:90% 00:36:29.950 --> 00:36:34.020 align:middle line:84% Let's look at a concrete example. 00:36:34.020 --> 00:36:38.280 align:middle line:84% So let's keep trying to look at 5. 00:36:38.280 --> 00:36:39.905 align:middle line:90% Let's say b minus a is also 5. 00:36:39.905 --> 00:36:43.950 align:middle line:90% 00:36:43.950 --> 00:36:45.430 align:middle line:90% Is this a true statement? 00:36:45.430 --> 00:37:00.820 align:middle line:90% 00:37:00.820 --> 00:37:07.540 align:middle line:84% Sorry, did somebody-- is this true for all c? 00:37:07.540 --> 00:37:15.160 align:middle line:90% 00:37:15.160 --> 00:37:17.590 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:37:17.590 --> 00:37:19.617 align:middle line:84% BRYNMOR CHAPMAN: So I claim that minus 1 00:37:19.617 --> 00:37:20.700 align:middle line:90% is not going to work here. 00:37:20.700 --> 00:37:26.384 align:middle line:90% 00:37:26.384 --> 00:37:28.270 align:middle line:90% Yeah? 00:37:28.270 --> 00:37:30.803 align:middle line:84% AUDIENCE: Anything to the 0 power is 1 is true. 00:37:30.803 --> 00:37:31.720 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:37:31.720 --> 00:37:34.220 align:middle line:84% So anything to the 0-th power is 1. 00:37:34.220 --> 00:37:36.280 align:middle line:90% That's fine. 00:37:36.280 --> 00:37:44.085 align:middle line:84% The problem is we're subtracting a from both of these exponents. 00:37:44.085 --> 00:37:47.740 align:middle line:84% So if, say, you have b equals 5 plus a, 00:37:47.740 --> 00:37:50.655 align:middle line:84% you're going to end up with 5 on the right-hand side. 00:37:50.655 --> 00:37:56.944 align:middle line:90% 00:37:56.944 --> 00:37:59.390 align:middle line:84% So our counterexample is going to be-- 00:37:59.390 --> 00:38:08.190 align:middle line:84% maybe we could have something like c to the 1 and c to the 6. 00:38:08.190 --> 00:38:11.380 align:middle line:84% And in particular, we're going to want c to be minus 1. 00:38:11.380 --> 00:38:27.425 align:middle line:90% 00:38:27.425 --> 00:38:29.138 align:middle line:90% Ah, n was five-- 00:38:29.138 --> 00:38:29.680 align:middle line:90% that's right. 00:38:29.680 --> 00:38:37.552 align:middle line:90% 00:38:37.552 --> 00:38:39.450 align:middle line:90% So this is our counterexample. 00:38:39.450 --> 00:38:43.410 align:middle line:84% We've exhibited a specific a, b, and n, 00:38:43.410 --> 00:38:48.860 align:middle line:84% where our purported theorem statement fails. 00:38:48.860 --> 00:38:52.340 align:middle line:84% It is not the case that minus 1 to the 1 00:38:52.340 --> 00:38:56.110 align:middle line:84% is congruent mod 5 to minus 1 to the 6. 00:38:56.110 --> 00:38:57.400 align:middle line:90% The left-hand side is minus 1. 00:38:57.400 --> 00:38:58.817 align:middle line:84% The right-hand side is positive 1. 00:38:58.817 --> 00:39:04.020 align:middle line:90% 00:39:04.020 --> 00:39:07.440 align:middle line:84% So when exponentiating, we have to be careful. 00:39:07.440 --> 00:39:10.900 align:middle line:84% We are allowed to replace the bases of exponents. 00:39:10.900 --> 00:39:13.196 align:middle line:84% We're not allowed to replace the exponents themselves. 00:39:13.196 --> 00:39:20.120 align:middle line:90% 00:39:20.120 --> 00:39:24.990 align:middle line:84% So we will see in a little while how we can get around 00:39:24.990 --> 00:39:26.200 align:middle line:90% this to some degree. 00:39:26.200 --> 00:39:28.060 align:middle line:84% We can do something with the exponents. 00:39:28.060 --> 00:39:29.405 align:middle line:90% It's just not this. 00:39:29.405 --> 00:39:37.530 align:middle line:90% 00:39:37.530 --> 00:39:40.140 align:middle line:84% Well, why don't we take a look at an example first. 00:39:40.140 --> 00:39:56.230 align:middle line:84% So suppose x is 11335 to the 11111 multiplied by 6 plus 7799 00:39:56.230 --> 00:40:03.760 align:middle line:90% to the 5,000. 00:40:03.760 --> 00:40:07.270 align:middle line:84% How might we compute the remainder 00:40:07.270 --> 00:40:09.810 align:middle line:90% of x after we divide by 100? 00:40:09.810 --> 00:40:14.530 align:middle line:90% 00:40:14.530 --> 00:40:18.130 align:middle line:84% How can we use the two terms that we did prove 00:40:18.130 --> 00:40:20.360 align:middle line:90% to compute that efficiently? 00:40:20.360 --> 00:40:27.850 align:middle line:90% 00:40:27.850 --> 00:40:29.250 align:middle line:90% What might our first step be? 00:40:29.250 --> 00:40:36.014 align:middle line:90% 00:40:36.014 --> 00:40:37.010 align:middle line:90% Mm-hmm? 00:40:37.010 --> 00:40:44.050 align:middle line:84% AUDIENCE: We try to distribute, separate out the 1,000 times 6 00:40:44.050 --> 00:40:48.545 align:middle line:90% plus 11,000 times the 7,000? 00:40:48.545 --> 00:40:50.420 align:middle line:84% BRYNMOR CHAPMAN: So the answer was distribute 00:40:50.420 --> 00:40:53.378 align:middle line:90% this across this addition. 00:40:53.378 --> 00:40:55.670 align:middle line:84% I claim that that's going to make our lives harder, not 00:40:55.670 --> 00:40:56.170 align:middle line:90% easier. 00:40:56.170 --> 00:41:03.560 align:middle line:90% 00:41:03.560 --> 00:41:08.360 align:middle line:84% So what was the theme of the two theorems that we just proved? 00:41:08.360 --> 00:41:11.570 align:middle line:84% If we want to do computations modulo n, 00:41:11.570 --> 00:41:15.950 align:middle line:84% we can replace things by other numbers 00:41:15.950 --> 00:41:20.400 align:middle line:84% that are easier to work with and which are congruent. 00:41:20.400 --> 00:41:22.975 align:middle line:90% 00:41:22.975 --> 00:41:28.100 align:middle line:84% So computing the remainder when we divide by 100, 00:41:28.100 --> 00:41:31.280 align:middle line:84% that's looking for a computation modulo 100. 00:41:31.280 --> 00:41:35.900 align:middle line:84% So we can replace these values, modulo 100, 00:41:35.900 --> 00:41:37.450 align:middle line:84% and make them easier to work with. 00:41:37.450 --> 00:41:40.040 align:middle line:90% 00:41:40.040 --> 00:41:43.790 align:middle line:84% Are any of these values that I've written down congruent 00:41:43.790 --> 00:41:45.250 align:middle line:90% to anything nice modulo 100? 00:41:45.250 --> 00:42:00.840 align:middle line:90% 00:42:00.840 --> 00:42:02.790 align:middle line:90% Anyone? 00:42:02.790 --> 00:42:03.900 align:middle line:90% Yeah? 00:42:03.900 --> 00:42:07.770 align:middle line:84% AUDIENCE: Is the first term that you're multiplying the 11,000-- 00:42:07.770 --> 00:42:12.690 align:middle line:90% 11,335 to the 11,000 whatever? 00:42:12.690 --> 00:42:16.033 align:middle line:84% Can you write that is congruent to 35 to the-- 00:42:16.033 --> 00:42:16.950 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:42:16.950 --> 00:42:18.210 align:middle line:90% AUDIENCE: 11,000 whatever. 00:42:18.210 --> 00:42:21.012 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so we can just get rid of these. 00:42:21.012 --> 00:42:23.880 align:middle line:90% That's congruent to 35. 00:42:23.880 --> 00:42:25.980 align:middle line:90% What else can we do? 00:42:25.980 --> 00:42:26.480 align:middle line:90% Yeah? 00:42:26.480 --> 00:42:27.938 align:middle line:84% AUDIENCE: Can you explain again why 00:42:27.938 --> 00:42:29.610 align:middle line:90% that-- we just get rid of that? 00:42:29.610 --> 00:42:32.290 align:middle line:84% BRYNMOR CHAPMAN: So we're computing modulo 100. 00:42:32.290 --> 00:42:36.120 align:middle line:84% We're looking for the remainder when we divide by 100. 00:42:36.120 --> 00:42:40.490 align:middle line:90% So we know that 1-- 00:42:40.490 --> 00:42:48.912 align:middle line:84% whoops, 11335 is congruent to 35 modulo 100. 00:42:48.912 --> 00:42:50.870 align:middle line:84% AUDIENCE: [INAUDIBLE] over the last two digits? 00:42:50.870 --> 00:42:51.787 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:42:51.787 --> 00:42:54.975 align:middle line:84% So we're getting rid of the first three digits 00:42:54.975 --> 00:42:56.350 align:middle line:84% and only looking at the remainder 00:42:56.350 --> 00:42:57.880 align:middle line:90% after we divide by 100. 00:42:57.880 --> 00:43:02.740 align:middle line:84% So basically, the idea is that we can do this division 00:43:02.740 --> 00:43:05.050 align:middle line:90% at intermediate steps. 00:43:05.050 --> 00:43:09.110 align:middle line:84% We can take remainders in the middle of our computation. 00:43:09.110 --> 00:43:11.595 align:middle line:84% That's what these two theorems are telling us. 00:43:11.595 --> 00:43:14.092 align:middle line:90% 00:43:14.092 --> 00:43:15.550 align:middle line:84% Anything else that we can simplify? 00:43:15.550 --> 00:43:21.830 align:middle line:90% 00:43:21.830 --> 00:43:22.330 align:middle line:90% No? 00:43:22.330 --> 00:43:25.930 align:middle line:84% Does somebody want to do that computation by hand? 00:43:25.930 --> 00:43:26.480 align:middle line:90% No, nobody? 00:43:26.480 --> 00:43:26.980 align:middle line:90% Yeah. 00:43:26.980 --> 00:43:27.932 align:middle line:90% OK. 00:43:27.932 --> 00:43:29.932 align:middle line:84% AUDIENCE: So we do the same thing with the 7,000 00:43:29.932 --> 00:43:31.067 align:middle line:90% [INAUDIBLE]. 00:43:31.067 --> 00:43:31.900 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:43:31.900 --> 00:43:32.620 align:middle line:90% AUDIENCE: It'd be 99. 00:43:32.620 --> 00:43:34.580 align:middle line:84% BRYNMOR CHAPMAN: OK, yeah, we can do the same thing here, 00:43:34.580 --> 00:43:35.745 align:middle line:90% turn that into just 99. 00:43:35.745 --> 00:43:39.280 align:middle line:90% 00:43:39.280 --> 00:43:41.175 align:middle line:84% I claim that we can take that a step further. 00:43:41.175 --> 00:43:45.130 align:middle line:90% 00:43:45.130 --> 00:43:47.980 align:middle line:84% 99 is not particularly easy to work with. 00:43:47.980 --> 00:43:49.730 align:middle line:84% I claim that there's something else that's 00:43:49.730 --> 00:43:51.950 align:middle line:90% equivalent to that that is. 00:43:51.950 --> 00:43:52.940 align:middle line:90% Yeah? 00:43:52.940 --> 00:43:54.230 align:middle line:90% AUDIENCE: 0? 00:43:54.230 --> 00:43:55.250 align:middle line:90% BRYNMOR CHAPMAN: Not 0. 00:43:55.250 --> 00:43:56.083 align:middle line:90% AUDIENCE: Negative 1 00:43:56.083 --> 00:43:57.950 align:middle line:84% BRYNMOR CHAPMAN: Negative 1, yeah. 00:43:57.950 --> 00:44:08.120 align:middle line:84% So we can replace this 7799 with negative 1. 00:44:08.120 --> 00:44:09.490 align:middle line:90% Negative 1 is easy to work with. 00:44:09.490 --> 00:44:12.374 align:middle line:90% 00:44:12.374 --> 00:44:13.525 align:middle line:90% So now what happens? 00:44:13.525 --> 00:44:23.330 align:middle line:90% 00:44:23.330 --> 00:44:24.470 align:middle line:90% Yeah? 00:44:24.470 --> 00:44:27.313 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:44:27.313 --> 00:44:28.230 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:44:28.230 --> 00:44:31.730 align:middle line:84% So let's exponentiate this negative 1. 00:44:31.730 --> 00:44:34.675 align:middle line:84% So we end up with 6 plus 1, which is 7. 00:44:34.675 --> 00:44:40.580 align:middle line:90% 00:44:40.580 --> 00:44:49.090 align:middle line:84% Now what do we do about 35 to the 11,111? 00:44:49.090 --> 00:44:51.490 align:middle line:84% Well, we don't really know how to do anything 00:44:51.490 --> 00:44:54.730 align:middle line:90% with exponents, right? 00:44:54.730 --> 00:44:59.590 align:middle line:84% So let's just try to start exponentiating it 00:44:59.590 --> 00:45:00.615 align:middle line:90% and see what happens. 00:45:00.615 --> 00:45:03.538 align:middle line:90% 00:45:03.538 --> 00:45:10.050 align:middle line:90% So what's 35 squared? 00:45:10.050 --> 00:45:16.690 align:middle line:90% 00:45:16.690 --> 00:45:18.160 align:middle line:90% Well, that's going to be-- 00:45:18.160 --> 00:45:23.260 align:middle line:84% well, if we actually compute it, it'll be 1,225, I think. 00:45:23.260 --> 00:45:25.030 align:middle line:90% Is that right? 00:45:25.030 --> 00:45:29.560 align:middle line:90% 1,175? 00:45:29.560 --> 00:45:31.412 align:middle line:90% 1,225. 00:45:31.412 --> 00:45:33.663 align:middle line:84% So that's going to be congruent to 25. 00:45:33.663 --> 00:45:40.300 align:middle line:90% 00:45:40.300 --> 00:45:41.400 align:middle line:90% 35 cubed? 00:45:41.400 --> 00:45:47.350 align:middle line:90% 00:45:47.350 --> 00:45:51.745 align:middle line:84% Well, it's going to be 25 times 35. 00:45:51.745 --> 00:45:53.870 align:middle line:90% So that's 875. 00:45:53.870 --> 00:45:55.800 align:middle line:84% So that's going to be congruent to 75. 00:45:55.800 --> 00:46:01.000 align:middle line:90% 00:46:01.000 --> 00:46:02.145 align:middle line:90% 35 to the fourth? 00:46:02.145 --> 00:46:09.730 align:middle line:90% 00:46:09.730 --> 00:46:13.520 align:middle line:84% Well, 35 to the fourth is going to be 35 squared, squared again, 00:46:13.520 --> 00:46:16.810 align:middle line:90% so 25 squared. 00:46:16.810 --> 00:46:18.495 align:middle line:90% 25 squared is 625. 00:46:18.495 --> 00:46:20.175 align:middle line:84% So that also has a remainder of 25. 00:46:20.175 --> 00:46:25.090 align:middle line:90% 00:46:25.090 --> 00:46:28.150 align:middle line:84% So what's going to happen after this point? 00:46:28.150 --> 00:46:30.585 align:middle line:84% What happens is we continue multiplying by 35? 00:46:30.585 --> 00:46:36.474 align:middle line:90% 00:46:36.474 --> 00:46:37.462 align:middle line:90% Mm-hmm? 00:46:37.462 --> 00:46:39.940 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:46:39.940 --> 00:46:40.980 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:46:40.980 --> 00:46:45.120 align:middle line:84% The answer was it'll alternate between 75 and 25. 00:46:45.120 --> 00:46:50.138 align:middle line:84% Whenever we have an odd power, as long as it's greater than 1, 00:46:50.138 --> 00:46:53.590 align:middle line:84% when we exponentiate 35, it's going to be 75. 00:46:53.590 --> 00:46:56.640 align:middle line:84% Whenever we have an even power, it's going to be 25. 00:46:56.640 --> 00:47:00.840 align:middle line:84% We can prove that formally by induction. 00:47:00.840 --> 00:47:02.820 align:middle line:90% I'm not going to do that here. 00:47:02.820 --> 00:47:06.610 align:middle line:84% But the point is 35, we've got an odd exponent here. 00:47:06.610 --> 00:47:12.335 align:middle line:90% So we can replace this with 75. 00:47:12.335 --> 00:47:18.488 align:middle line:90% 00:47:18.488 --> 00:47:23.195 align:middle line:84% And now 75 times 7 is going to have a remainder of 25. 00:47:23.195 --> 00:47:32.065 align:middle line:90% 00:47:32.065 --> 00:47:34.380 align:middle line:84% So does that make sense to everybody? 00:47:34.380 --> 00:47:38.040 align:middle line:84% We perform the computation, but instead of just doing it 00:47:38.040 --> 00:47:41.000 align:middle line:84% over the integers and then reducing mod 100, 00:47:41.000 --> 00:47:44.870 align:middle line:84% we're reducing mod 100 whenever we can. 00:47:44.870 --> 00:47:48.520 align:middle line:84% We're always trying to replace intermediate values with things 00:47:48.520 --> 00:47:51.500 align:middle line:90% that are easier to work with. 00:47:51.500 --> 00:47:53.900 align:middle line:84% And that simplifies our computation considerably. 00:47:53.900 --> 00:47:54.490 align:middle line:90% Yeah? 00:47:54.490 --> 00:48:00.250 align:middle line:84% AUDIENCE: How did you know 75 is [INAUDIBLE] 25? 00:48:00.250 --> 00:48:02.320 align:middle line:84% BRYNMOR CHAPMAN: You can perform the computation. 00:48:02.320 --> 00:48:04.890 align:middle line:84% 25 by 21 is possibly an easier one to do. 00:48:04.890 --> 00:48:17.770 align:middle line:90% 00:48:17.770 --> 00:48:20.500 align:middle line:90% Any other questions? 00:48:20.500 --> 00:48:21.640 align:middle line:90% Yeah? 00:48:21.640 --> 00:48:25.600 align:middle line:84% AUDIENCE: I think there was a step where we went from 6 plus 00:48:25.600 --> 00:48:27.905 align:middle line:90% negative 1 to the 5k to 7. 00:48:27.905 --> 00:48:28.780 align:middle line:90% BRYNMOR CHAPMAN: Yes. 00:48:28.780 --> 00:48:31.180 align:middle line:84% AUDIENCE: I was just wondering what happened there. 00:48:31.180 --> 00:48:38.831 align:middle line:84% BRYNMOR CHAPMAN: So we had 6 plus negative 1 to the 5k. 00:48:38.831 --> 00:48:42.660 align:middle line:84% So we are exponentiating this negative 1 over the integers. 00:48:42.660 --> 00:48:46.315 align:middle line:84% Negative 1 to any even power is going to be 1. 00:48:46.315 --> 00:48:50.250 align:middle line:90% 00:48:50.250 --> 00:48:51.292 align:middle line:90% AUDIENCE: k is even? 00:48:51.292 --> 00:48:52.250 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 00:48:52.250 --> 00:48:53.540 align:middle line:90% AUDIENCE: Do we know k's even? 00:48:53.540 --> 00:48:57.920 align:middle line:84% BRYNMOR CHAPMAN: Oh, sorry, 5k, 5,000. 00:48:57.920 --> 00:49:01.040 align:middle line:90% Sorry, it's not actually-- 00:49:01.040 --> 00:49:05.830 align:middle line:90% yeah, my bad, this kind of k. 00:49:05.830 --> 00:49:13.580 align:middle line:90% 00:49:13.580 --> 00:49:14.280 align:middle line:90% Sorry. 00:49:14.280 --> 00:49:15.450 align:middle line:90% That's my bad. 00:49:15.450 --> 00:49:16.650 align:middle line:90% Any other questions? 00:49:16.650 --> 00:49:20.240 align:middle line:90% 00:49:20.240 --> 00:49:21.200 align:middle line:90% Yeah? 00:49:21.200 --> 00:49:22.354 align:middle line:90% AUDIENCE: So [INAUDIBLE]. 00:49:22.354 --> 00:49:26.200 align:middle line:90% 00:49:26.200 --> 00:49:27.200 align:middle line:90% BRYNMOR CHAPMAN: Pardon? 00:49:27.200 --> 00:49:29.900 align:middle line:84% AUDIENCE: Will all phases exhibit a pattern? 00:49:29.900 --> 00:49:30.960 align:middle line:90% BRYNMOR CHAPMAN: Oh. 00:49:30.960 --> 00:49:32.960 align:middle line:84% So the question was, is there always 00:49:32.960 --> 00:49:34.345 align:middle line:90% going to be a pattern like this? 00:49:34.345 --> 00:49:37.200 align:middle line:90% 00:49:37.200 --> 00:49:40.690 align:middle line:84% So this is kind of touching on stuff that we will see later. 00:49:40.690 --> 00:49:42.760 align:middle line:84% But yes, there will always be some pattern. 00:49:42.760 --> 00:49:46.000 align:middle line:90% It may not be this easy to find. 00:49:46.000 --> 00:49:49.830 align:middle line:84% But the basic idea is that there are only 100 different values 00:49:49.830 --> 00:49:53.400 align:middle line:90% that it could take, modulo 100. 00:49:53.400 --> 00:49:59.170 align:middle line:84% So if you have more than 100 powers, 00:49:59.170 --> 00:50:02.430 align:middle line:84% something's going to have to collide. 00:50:02.430 --> 00:50:06.420 align:middle line:84% And At that point, it'll start repeating. 00:50:06.420 --> 00:50:09.490 align:middle line:84% Here, it happened to be quite nice. 00:50:09.490 --> 00:50:12.130 align:middle line:84% It started repeating very quickly. 00:50:12.130 --> 00:50:14.340 align:middle line:90% And the period is only two. 00:50:14.340 --> 00:50:17.760 align:middle line:84% That may not always be the case, though. 00:50:17.760 --> 00:50:23.880 align:middle line:84% but we will see, in a moment, something more 00:50:23.880 --> 00:50:26.200 align:middle line:84% about what we can say about that. 00:50:26.200 --> 00:50:35.200 align:middle line:90% 00:50:35.200 --> 00:50:38.590 align:middle line:84% So we've seen how to add, subtract, multiply, 00:50:38.590 --> 00:50:40.190 align:middle line:84% how to change the bases of exponents. 00:50:40.190 --> 00:50:43.150 align:middle line:90% What about division? 00:50:43.150 --> 00:50:45.860 align:middle line:84% Are we allowed to divide modulo n? 00:50:45.860 --> 00:50:54.955 align:middle line:84% So suppose we have 3x is congruent to 3 modulo 6. 00:50:54.955 --> 00:50:58.540 align:middle line:90% 00:50:58.540 --> 00:51:00.790 align:middle line:84% So if we had this over the integers, 00:51:00.790 --> 00:51:03.400 align:middle line:90% say, or over the reals-- 00:51:03.400 --> 00:51:05.570 align:middle line:84% I guess reals are a better example-- 00:51:05.570 --> 00:51:11.580 align:middle line:84% if we just had 3x equals 3, how would we solve this for x? 00:51:11.580 --> 00:51:15.760 align:middle line:90% 00:51:15.760 --> 00:51:17.670 align:middle line:84% It's not a trick question, I promise. 00:51:17.670 --> 00:51:20.560 align:middle line:90% 00:51:20.560 --> 00:51:21.169 align:middle line:90% Yeah? 00:51:21.169 --> 00:51:22.086 align:middle line:90% AUDIENCE: Divide by 3. 00:51:22.086 --> 00:51:24.380 align:middle line:84% BRYNMOR CHAPMAN: Yeah, you just divide both sides by 3. 00:51:24.380 --> 00:51:29.650 align:middle line:84% You can cancel off this 3, and you just get x equals 1. 00:51:29.650 --> 00:51:33.360 align:middle line:84% So now the actual question is, can we do that modulo n? 00:51:33.360 --> 00:51:34.960 align:middle line:90% Can we do that modulo 6? 00:51:34.960 --> 00:51:41.370 align:middle line:84% Are we allowed to just cancel these 3's and conclude that-- 00:51:41.370 --> 00:51:46.410 align:middle line:84% oops-- x is congruent to 1 modulo 6? 00:51:46.410 --> 00:51:46.995 align:middle line:90% Yeah? 00:51:46.995 --> 00:51:49.620 align:middle line:84% AUDIENCE: I think so because the [INAUDIBLE] multiplying by 1/3 00:51:49.620 --> 00:51:50.367 align:middle line:90% [INAUDIBLE]. 00:51:50.367 --> 00:51:51.200 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:51:51.200 --> 00:51:53.658 align:middle line:84% So the answer was yes, because it's the same as multiplying 00:51:53.658 --> 00:51:54.960 align:middle line:90% by 1/3. 00:51:54.960 --> 00:52:01.500 align:middle line:84% So remember that 1/3 is not really something that we can-- 00:52:01.500 --> 00:52:06.780 align:middle line:84% it doesn't really make sense modulo, modulo n, or at least, 00:52:06.780 --> 00:52:08.140 align:middle line:90% not modulo 6. 00:52:08.140 --> 00:52:10.062 align:middle line:90% Sometimes it might make sense. 00:52:10.062 --> 00:52:11.020 align:middle line:90% Sometimes it might not. 00:52:11.020 --> 00:52:14.340 align:middle line:84% We'll see more about that in a moment. 00:52:14.340 --> 00:52:16.075 align:middle line:90% But what is a third? 00:52:16.075 --> 00:52:17.283 align:middle line:90% We're talking about integers. 00:52:17.283 --> 00:52:23.640 align:middle line:90% 00:52:23.640 --> 00:52:26.250 align:middle line:90% Yeah? 00:52:26.250 --> 00:52:28.663 align:middle line:90% Sorry? 00:52:28.663 --> 00:52:30.106 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:52:30.106 --> 00:52:37.023 align:middle line:90% 00:52:37.023 --> 00:52:38.940 align:middle line:84% BRYNMOR CHAPMAN: So other way around, multiply 00:52:38.940 --> 00:52:40.120 align:middle line:90% instead of dividing? 00:52:40.120 --> 00:52:41.790 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:52:41.790 --> 00:52:43.650 align:middle line:90% BRYNMOR CHAPMAN: OK. 00:52:43.650 --> 00:52:48.460 align:middle line:84% So the suggestion was instead of dividing by 3, 00:52:48.460 --> 00:52:50.760 align:middle line:84% let's try multiplying by something, 00:52:50.760 --> 00:52:54.060 align:middle line:90% so that we get 1 here. 00:52:54.060 --> 00:53:01.860 align:middle line:84% So let's try to multiply this, multiply both sides by some a. 00:53:01.860 --> 00:53:05.070 align:middle line:84% And this should end up being 1, if we choose our 00:53:05.070 --> 00:53:08.700 align:middle line:90% a appropriately. 00:53:08.700 --> 00:53:10.040 align:middle line:90% What a do we choose? 00:53:10.040 --> 00:53:14.040 align:middle line:90% 00:53:14.040 --> 00:53:15.815 align:middle line:90% Does there exist such an a? 00:53:15.815 --> 00:53:20.130 align:middle line:90% 00:53:20.130 --> 00:53:21.150 align:middle line:90% Yeah? 00:53:21.150 --> 00:53:25.650 align:middle line:84% AUDIENCE: The original claim is false because if we set x equal 00:53:25.650 --> 00:53:29.620 align:middle line:90% to 3, 3x is [INAUDIBLE]. 00:53:29.620 --> 00:53:31.310 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:53:31.310 --> 00:53:33.790 align:middle line:90% So the original claim is false. 00:53:33.790 --> 00:53:38.310 align:middle line:84% The counterexample is that we could take x's 3 here. 00:53:38.310 --> 00:53:40.670 align:middle line:90% 3 times 3 is 9. 00:53:40.670 --> 00:53:44.320 align:middle line:90% That is congruent to 3. 00:53:44.320 --> 00:53:45.970 align:middle line:90% But x is not congruent to 1. 00:53:45.970 --> 00:53:48.940 align:middle line:90% 00:53:48.940 --> 00:53:54.520 align:middle line:84% So the answer, I guess, to the followup question 00:53:54.520 --> 00:53:58.310 align:middle line:90% is that no, there is no such a. 00:53:58.310 --> 00:54:03.865 align:middle line:84% We cannot divide by 3 or multiply by something to get 1. 00:54:03.865 --> 00:54:06.520 align:middle line:90% 00:54:06.520 --> 00:54:09.040 align:middle line:84% And this is basically the same thing 00:54:09.040 --> 00:54:10.340 align:middle line:90% that we were doing last week. 00:54:10.340 --> 00:54:20.150 align:middle line:84% We're looking for a solution to 3a is congruent to 1 modulo 6. 00:54:20.150 --> 00:54:27.520 align:middle line:84% If we unpack what this actually means, 6 divides 3a minus 1. 00:54:27.520 --> 00:54:32.300 align:middle line:90% 00:54:32.300 --> 00:54:33.670 align:middle line:90% Is there a solution to this? 00:54:33.670 --> 00:54:37.005 align:middle line:90% 00:54:37.005 --> 00:54:37.505 align:middle line:90% No. 00:54:37.505 --> 00:54:47.750 align:middle line:90% 00:54:47.750 --> 00:54:49.430 align:middle line:84% So basically, that's asking if we 00:54:49.430 --> 00:54:54.700 align:middle line:84% can write 1 as a linear combination of 6 and 3. 00:54:54.700 --> 00:54:57.368 align:middle line:90% 00:54:57.368 --> 00:55:00.025 align:middle line:84% But 1 is not a multiple of their gcd, so we cannot. 00:55:00.025 --> 00:55:06.590 align:middle line:90% 00:55:06.590 --> 00:55:10.280 align:middle line:84% So does that make sense to everybody? 00:55:10.280 --> 00:55:14.540 align:middle line:90% So we cannot always divide. 00:55:14.540 --> 00:55:17.005 align:middle line:84% Are there circumstances under which we can divide? 00:55:17.005 --> 00:55:28.705 align:middle line:90% 00:55:28.705 --> 00:55:29.205 align:middle line:90% Yeah? 00:55:29.205 --> 00:55:33.510 align:middle line:90% 00:55:33.510 --> 00:55:38.820 align:middle line:84% AUDIENCE: Oh, I was thinking you divide the 6, as well? 00:55:38.820 --> 00:55:40.320 align:middle line:84% BRYNMOR CHAPMAN: Oh, OK, so the idea 00:55:40.320 --> 00:55:44.350 align:middle line:90% was to divide the 6 as well. 00:55:44.350 --> 00:55:46.170 align:middle line:84% Yeah, I suppose you could do that. 00:55:46.170 --> 00:55:51.250 align:middle line:84% But we're specifically trying to do arithmetic modulo 6. 00:55:51.250 --> 00:55:53.660 align:middle line:84% So that may or may not make sense. 00:55:53.660 --> 00:55:59.445 align:middle line:90% 00:55:59.445 --> 00:56:07.300 align:middle line:84% But I mean, yeah, I suppose you could say 6 divides 3x minus 3a. 00:56:07.300 --> 00:56:13.830 align:middle line:84% And so you could factor out a 3 and say that x is at least odd. 00:56:13.830 --> 00:56:16.920 align:middle line:84% And in fact, that does characterize the solutions 00:56:16.920 --> 00:56:19.030 align:middle line:90% to the original problem. 00:56:19.030 --> 00:56:22.268 align:middle line:84% It's just that may or may not be what 00:56:22.268 --> 00:56:23.435 align:middle line:90% you're actually looking for. 00:56:23.435 --> 00:56:31.460 align:middle line:90% 00:56:31.460 --> 00:56:34.275 align:middle line:84% So under what circumstances are we allowed to divide? 00:56:34.275 --> 00:56:41.860 align:middle line:90% 00:56:41.860 --> 00:56:45.430 align:middle line:84% So we just saw that we can't divide here, because there's 00:56:45.430 --> 00:56:54.640 align:middle line:90% no solution to this equation. 00:56:54.640 --> 00:56:56.770 align:middle line:90% What if there is a solution? 00:56:56.770 --> 00:57:02.983 align:middle line:84% What if instead of modulo 6, we had modulo 5? 00:57:02.983 --> 00:57:13.330 align:middle line:90% 00:57:13.330 --> 00:57:14.850 align:middle line:84% Would we be allowed to divide here? 00:57:14.850 --> 00:57:23.420 align:middle line:90% 00:57:23.420 --> 00:57:25.830 align:middle line:84% If we have 3x is congruent to 3 modulo 5, 00:57:25.830 --> 00:57:30.640 align:middle line:84% can we then conclude x is congruent to 1 modulo 5? 00:57:30.640 --> 00:57:33.260 align:middle line:90% 00:57:33.260 --> 00:57:34.425 align:middle line:90% Seeing some nods. 00:57:34.425 --> 00:57:34.925 align:middle line:90% Why? 00:57:34.925 --> 00:57:41.810 align:middle line:90% 00:57:41.810 --> 00:57:43.770 align:middle line:84% Well, basically, exactly the same reason, 00:57:43.770 --> 00:57:50.580 align:middle line:84% this exactly characterizes when what we described will work. 00:57:50.580 --> 00:57:56.570 align:middle line:84% If the modulus and the coefficient 00:57:56.570 --> 00:58:02.750 align:middle line:84% we're trying to divide by are co-prime, if their gcd is 1, 00:58:02.750 --> 00:58:04.030 align:middle line:90% then we'll be able to divide. 00:58:04.030 --> 00:58:09.480 align:middle line:90% 00:58:09.480 --> 00:58:13.460 align:middle line:84% So more formally, let's write out what we mean. 00:58:13.460 --> 00:58:16.700 align:middle line:84% So let's start with the definition. 00:58:16.700 --> 00:58:32.700 align:middle line:84% So we'll say that a multiplicative inverse of-- 00:58:32.700 --> 00:58:35.100 align:middle line:90% let's call it x-- 00:58:35.100 --> 00:58:50.880 align:middle line:84% modulo n, which we will denote x to the negative 1 00:58:50.880 --> 00:58:57.460 align:middle line:84% is a number which, as you might expect, 00:58:57.460 --> 00:58:59.636 align:middle line:84% satisfies the following congruence. 00:58:59.636 --> 00:59:08.760 align:middle line:90% 00:59:08.760 --> 00:59:11.670 align:middle line:84% So just as we have multiplicative inverses 00:59:11.670 --> 00:59:12.940 align:middle line:90% over the reals-- 00:59:12.940 --> 00:59:17.110 align:middle line:84% so x times x inverse should be equal to 1, in that case-- 00:59:17.110 --> 00:59:19.850 align:middle line:84% we are defining something analogous 00:59:19.850 --> 00:59:21.145 align:middle line:90% when we're working mod n. 00:59:21.145 --> 00:59:24.275 align:middle line:90% 00:59:24.275 --> 00:59:26.245 align:middle line:84% So does that definition make sense to people? 00:59:26.245 --> 00:59:31.010 align:middle line:90% 00:59:31.010 --> 00:59:40.880 align:middle line:84% So as an example, 1 inverse, so 1 00:59:40.880 --> 00:59:43.220 align:middle line:84% is a multiplicative inverse of 1. 00:59:43.220 --> 00:59:57.213 align:middle line:84% Maybe I should write it like that, mod anything. 00:59:57.213 --> 01:00:01.100 align:middle line:90% 01:00:01.100 --> 01:00:15.050 align:middle line:84% 2 is a multiplicative inverse of 3 mod 5 because 2 times 3 is 6. 01:00:15.050 --> 01:00:18.280 align:middle line:90% 6 is congruent to 1 modulo 5. 01:00:18.280 --> 01:00:21.600 align:middle line:84% So 2 and 3 are modulo inverse, modulo inverses of each other, 01:00:21.600 --> 01:00:22.310 align:middle line:90% modulo 5. 01:00:22.310 --> 01:00:26.139 align:middle line:90% 01:00:26.139 --> 01:00:29.070 align:middle line:84% AUDIENCE: Can you explain that again, the last part of it? 01:00:29.070 --> 01:00:30.600 align:middle line:90% BRYNMOR CHAPMAN: The last part? 01:00:30.600 --> 01:00:31.380 align:middle line:90% Yeah. 01:00:31.380 --> 01:00:40.500 align:middle line:84% So what this is saying is that 2 times 3 01:00:40.500 --> 01:00:43.050 align:middle line:90% is congruent to 1 modulo-- 01:00:43.050 --> 01:00:44.555 align:middle line:84% oops, I should write it like this. 01:00:44.555 --> 01:00:49.255 align:middle line:90% 01:00:49.255 --> 01:00:52.740 align:middle line:90% So 2 times 3 is 6. 01:00:52.740 --> 01:00:56.410 align:middle line:90% 6 is congruent to 1 modulo 5. 01:00:56.410 --> 01:01:00.000 align:middle line:84% So that means that 2 and 3 are multiplicative inverses 01:01:00.000 --> 01:01:03.735 align:middle line:84% of each other modulo 5 because when you multiply them, 01:01:03.735 --> 01:01:04.235 align:middle line:90% you get 1. 01:01:04.235 --> 01:01:07.470 align:middle line:90% 01:01:07.470 --> 01:01:08.750 align:middle line:90% Does that make sense? 01:01:08.750 --> 01:01:21.880 align:middle line:90% 01:01:21.880 --> 01:01:24.780 align:middle line:84% So when do multiplicative inverses exist? 01:01:24.780 --> 01:01:45.130 align:middle line:90% 01:01:45.130 --> 01:01:46.975 align:middle line:84% Can anybody finish this theorem statement? 01:01:46.975 --> 01:01:48.225 align:middle line:90% Can anybody characterize this? 01:01:48.225 --> 01:01:53.710 align:middle line:90% 01:01:53.710 --> 01:01:55.930 align:middle line:90% When is a invertible modulo n? 01:01:55.930 --> 01:01:57.090 align:middle line:90% Yeah? 01:01:57.090 --> 01:01:59.290 align:middle line:90% AUDIENCE: Then you [INAUDIBLE]. 01:01:59.290 --> 01:02:06.180 align:middle line:84% BRYNMOR CHAPMAN: Yeah, when the gcd of a and n is 1. 01:02:06.180 --> 01:02:12.856 align:middle line:90% 01:02:12.856 --> 01:02:13.990 align:middle line:90% And why is that? 01:02:13.990 --> 01:02:19.460 align:middle line:90% 01:02:19.460 --> 01:02:23.720 align:middle line:84% AUDIENCE: It wasn't 1 [INAUDIBLE] 01:02:23.720 --> 01:02:28.790 align:middle line:84% multiply a by the numbers you could [INAUDIBLE]. 01:02:28.790 --> 01:02:37.220 align:middle line:90% 01:02:37.220 --> 01:02:40.980 align:middle line:84% BRYNMOR CHAPMAN: OK, yeah, so that's good intuition. 01:02:40.980 --> 01:02:44.000 align:middle line:84% If their gcd were not 1, like, say, 01:02:44.000 --> 01:02:49.423 align:middle line:84% the gcd is 2, so both a and n are even-- 01:02:49.423 --> 01:02:50.840 align:middle line:84% multiplying a by anything is going 01:02:50.840 --> 01:02:52.005 align:middle line:90% to give you an even number. 01:02:52.005 --> 01:02:53.630 align:middle line:84% That's never going to be congruent to 1 01:02:53.630 --> 01:02:55.075 align:middle line:90% modulo another even number. 01:02:55.075 --> 01:02:58.765 align:middle line:90% 01:02:58.765 --> 01:03:18.630 align:middle line:84% So more formally, so a has an inverse mod n if and only 01:03:18.630 --> 01:03:26.460 align:middle line:84% if there exists a b-- so this is just the definition-- 01:03:26.460 --> 01:03:31.040 align:middle line:84% such that ab is congruent mod n to 1. 01:03:31.040 --> 01:03:34.890 align:middle line:90% 01:03:34.890 --> 01:03:36.890 align:middle line:84% Now, how do we characterize when this is true? 01:03:36.890 --> 01:03:39.690 align:middle line:90% 01:03:39.690 --> 01:03:42.270 align:middle line:84% Well, let's unpack this definition here. 01:03:42.270 --> 01:03:45.106 align:middle line:84% So this means that n divides their difference. 01:03:45.106 --> 01:03:54.810 align:middle line:90% 01:03:54.810 --> 01:04:01.160 align:middle line:84% So-- oops-- why don't I write this as ab minus 1. 01:04:01.160 --> 01:04:11.060 align:middle line:90% 01:04:11.060 --> 01:04:14.530 align:middle line:84% So we're saying that n divides ab minus 1. 01:04:14.530 --> 01:04:17.290 align:middle line:84% We're looking at what that division actually looks like, 01:04:17.290 --> 01:04:20.720 align:middle line:90% what the factor of n is. 01:04:20.720 --> 01:04:23.935 align:middle line:84% So there's some q such that qn is ab minus 1. 01:04:23.935 --> 01:04:28.150 align:middle line:90% 01:04:28.150 --> 01:04:31.750 align:middle line:90% Now we can rewrite this. 01:04:31.750 --> 01:04:35.620 align:middle line:90% So 1 equals ab minus qn. 01:04:35.620 --> 01:04:39.510 align:middle line:90% 01:04:39.510 --> 01:04:42.730 align:middle line:84% So we're saying there exists b and q such that 1 01:04:42.730 --> 01:04:48.820 align:middle line:90% equals ab plus minus q times n. 01:04:48.820 --> 01:04:52.540 align:middle line:84% So this is just saying that 1 is an ILC of an. 01:04:52.540 --> 01:05:09.958 align:middle line:90% 01:05:09.958 --> 01:05:13.400 align:middle line:84% And as we saw before, this happens if and only 01:05:13.400 --> 01:05:14.455 align:middle line:90% if their gcd is 1. 01:05:14.455 --> 01:05:26.810 align:middle line:90% 01:05:26.810 --> 01:05:28.410 align:middle line:84% So does that make sense to everybody? 01:05:28.410 --> 01:05:39.560 align:middle line:90% 01:05:39.560 --> 01:05:43.370 align:middle line:84% So if two numbers have a gcd of 1, 01:05:43.370 --> 01:05:47.305 align:middle line:84% then we can divide by 1 when working modulo the other. 01:05:47.305 --> 01:05:55.662 align:middle line:90% 01:05:55.662 --> 01:06:06.560 align:middle line:84% So if we have, say, 7x is congruent mod 30 to 14, 01:06:06.560 --> 01:06:10.530 align:middle line:84% then we are allowed to divide by both sides because the gcd of 7 01:06:10.530 --> 01:06:12.220 align:middle line:90% and 30 is 1. 01:06:12.220 --> 01:06:15.000 align:middle line:84% So we're allowed to divide both sides by 7. 01:06:15.000 --> 01:06:20.015 align:middle line:84% So we are allowed to conclude that x is congruent mod 30 to 2. 01:06:20.015 --> 01:06:22.620 align:middle line:90% 01:06:22.620 --> 01:06:24.900 align:middle line:84% I suppose more formally, we want to find 01:06:24.900 --> 01:06:26.660 align:middle line:90% a multiplicative inverse of 7. 01:06:26.660 --> 01:06:29.778 align:middle line:90% 01:06:29.778 --> 01:06:35.350 align:middle line:84% And I think 13 is such an inverse, 7 by 1391. 01:06:35.350 --> 01:06:36.150 align:middle line:90% Yeah. 01:06:36.150 --> 01:06:40.540 align:middle line:84% So if we multiply both sides by negative 13, 01:06:40.540 --> 01:06:43.200 align:middle line:84% we'll get a coefficient of 1 over here, 01:06:43.200 --> 01:06:46.025 align:middle line:84% and we'll get 2 on the right-hand side. 01:06:46.025 --> 01:06:51.330 align:middle line:90% 01:06:51.330 --> 01:06:53.570 align:middle line:84% So sometimes, you can divide, but not always. 01:06:53.570 --> 01:07:20.660 align:middle line:90% 01:07:20.660 --> 01:07:26.440 align:middle line:84% So let's use that now to figure out what more we 01:07:26.440 --> 01:07:29.640 align:middle line:90% can say about exponentiation. 01:07:29.640 --> 01:07:33.820 align:middle line:84% so the following is a theorem, which is attributed 01:07:33.820 --> 01:07:35.510 align:middle line:90% to Fermat, as many things are. 01:07:35.510 --> 01:07:37.970 align:middle line:84% It's often known as Fermat's Little theorem, 01:07:37.970 --> 01:07:42.520 align:middle line:84% not to be confused with Fermat's Last theorem, which Fermat-- 01:07:42.520 --> 01:07:45.110 align:middle line:84% I feel like that one should not be attributed to him at all, 01:07:45.110 --> 01:07:46.670 align:middle line:90% because he didn't prove it. 01:07:46.670 --> 01:07:49.570 align:middle line:84% Or I think he thought he knew a proof 01:07:49.570 --> 01:07:52.002 align:middle line:84% and was like a complete fuck up, but whatever. 01:07:52.002 --> 01:07:55.025 align:middle line:90% 01:07:55.025 --> 01:07:56.900 align:middle line:84% Fermat's Little theorem is a different story. 01:07:56.900 --> 01:07:58.610 align:middle line:90% It is much more elementary. 01:07:58.610 --> 01:08:01.000 align:middle line:90% He did actually prove it. 01:08:01.000 --> 01:08:04.310 align:middle line:90% It states the following. 01:08:04.310 --> 01:08:14.490 align:middle line:84% So if p is prime, and a is not a multiple of p, 01:08:14.490 --> 01:08:24.170 align:middle line:84% so a not congruent mod p to 0, then a to the p minus 1 01:08:24.170 --> 01:08:27.535 align:middle line:90% is congruent mod p to 1. 01:08:27.535 --> 01:08:31.910 align:middle line:90% 01:08:31.910 --> 01:08:39.529 align:middle line:84% So if you think about what this is saying a little bit, 01:08:39.529 --> 01:08:41.029 align:middle line:84% it's similar to what we were trying 01:08:41.029 --> 01:08:43.550 align:middle line:90% to do with exponents earlier. 01:08:43.550 --> 01:08:48.800 align:middle line:84% But instead of getting rid of multiples of p in an exponent 01:08:48.800 --> 01:08:51.229 align:middle line:84% when we're working mod p, we're instead 01:08:51.229 --> 01:08:55.040 align:middle line:84% going to get rid of multiples of p minus 1. 01:08:55.040 --> 01:08:59.930 align:middle line:84% So basically, we can always factor out an a to the p minus 1 01:08:59.930 --> 01:09:05.406 align:middle line:84% without changing the end result. Yeah? 01:09:05.406 --> 01:09:06.735 align:middle line:90% AUDIENCE: [INAUDIBLE]? 01:09:06.735 --> 01:09:07.693 align:middle line:90% BRYNMOR CHAPMAN: Sorry. 01:09:07.693 --> 01:09:11.130 align:middle line:90% 01:09:11.130 --> 01:09:12.689 align:middle line:90% Thank you, Zach. 01:09:12.689 --> 01:09:15.000 align:middle line:90% I promise I'm awake. 01:09:15.000 --> 01:09:19.319 align:middle line:84% Yeah, if a is congruent to 0, then clearly, this theorem 01:09:19.319 --> 01:09:21.569 align:middle line:90% is false. 01:09:21.569 --> 01:09:23.410 align:middle line:90% a should not be congruent to 0. 01:09:23.410 --> 01:09:26.554 align:middle line:84% If a is congruent to 0, then a to the p minus 1 is also 0. 01:09:26.554 --> 01:09:30.916 align:middle line:90% 01:09:30.916 --> 01:09:32.500 align:middle line:90% So how could we prove this? 01:09:32.500 --> 01:09:33.080 align:middle line:90% Any ideas? 01:09:33.080 --> 01:09:43.740 align:middle line:90% 01:09:43.740 --> 01:09:46.180 align:middle line:84% It's a bit of a weird proof idea. 01:09:46.180 --> 01:09:50.310 align:middle line:84% Let's consider the following set. 01:09:50.310 --> 01:10:11.860 align:middle line:84% So a, 2a, 3a, all the way up to p minus 1 times a, now what can 01:10:11.860 --> 01:10:14.365 align:middle line:84% we say about this set if we reduce everything in it modulo 01:10:14.365 --> 01:10:14.865 align:middle line:90% p? 01:10:14.865 --> 01:10:27.395 align:middle line:90% 01:10:27.395 --> 01:10:28.619 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:10:28.619 --> 01:10:29.536 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 01:10:29.536 --> 01:10:32.050 align:middle line:90% 01:10:32.050 --> 01:10:35.920 align:middle line:84% So the answer was that this is basically equivalent to-- this 01:10:35.920 --> 01:10:37.350 align:middle line:90% is abusing notation a bit. 01:10:37.350 --> 01:10:46.675 align:middle line:90% 01:10:46.675 --> 01:10:50.372 align:middle line:84% The claim is that all of these are distinct modulo p, 01:10:50.372 --> 01:10:51.330 align:middle line:90% and none of them are 0. 01:10:51.330 --> 01:10:54.290 align:middle line:90% 01:10:54.290 --> 01:10:56.890 align:middle line:90% Why is that? 01:10:56.890 --> 01:11:00.408 align:middle line:84% Well, first, probably the easier part, why is none of these 0? 01:11:00.408 --> 01:11:02.950 align:middle line:84% That's basically saying, why is none of them a multiple of p? 01:11:02.950 --> 01:11:07.160 align:middle line:90% 01:11:07.160 --> 01:11:09.620 align:middle line:84% Well, all of the coefficients on a are less than p. 01:11:09.620 --> 01:11:13.670 align:middle line:84% So p doesn't divide any of those. 01:11:13.670 --> 01:11:16.520 align:middle line:84% p also doesn't divide a, by assumption. 01:11:16.520 --> 01:11:21.065 align:middle line:84% and p is prime, so it doesn't divide any of these products. 01:11:21.065 --> 01:11:24.260 align:middle line:90% So none of these is 0. 01:11:24.260 --> 01:11:26.380 align:middle line:84% Why are they all distinct modulo p? 01:11:26.380 --> 01:11:38.060 align:middle line:90% 01:11:38.060 --> 01:11:39.440 align:middle line:90% Yeah? 01:11:39.440 --> 01:11:44.340 align:middle line:84% AUDIENCE: If we think of a being a different system and modulo p, 01:11:44.340 --> 01:11:50.990 align:middle line:84% then they're all some constant from 1 to minus 1 times that n. 01:11:50.990 --> 01:11:56.150 align:middle line:84% And none of those will be the same or [INAUDIBLE]. 01:11:56.150 --> 01:11:57.140 align:middle line:90% BRYNMOR CHAPMAN: OK. 01:11:57.140 --> 01:11:58.490 align:middle line:90% Let me try and rephrase that. 01:11:58.490 --> 01:12:01.170 align:middle line:90% 01:12:01.170 --> 01:12:06.090 align:middle line:84% So if you look at two different values, 01:12:06.090 --> 01:12:11.490 align:middle line:84% like two different coefficients, you said, so 01:12:11.490 --> 01:12:17.800 align:middle line:84% suppose for contradiction, a times i equals a times j. 01:12:17.800 --> 01:12:20.685 align:middle line:84% So we're actually going to use a instead of n. 01:12:20.685 --> 01:12:23.770 align:middle line:90% 01:12:23.770 --> 01:12:27.570 align:middle line:84% Or sorry, this should be small p. 01:12:27.570 --> 01:12:30.420 align:middle line:90% 01:12:30.420 --> 01:12:31.640 align:middle line:90% What is this telling us? 01:12:31.640 --> 01:12:43.530 align:middle line:90% 01:12:43.530 --> 01:12:45.270 align:middle line:90% Yeah? 01:12:45.270 --> 01:12:48.300 align:middle line:84% AUDIENCE: So p prime and a and b are co-prime. 01:12:48.300 --> 01:12:49.550 align:middle line:90% BRYNMOR CHAPMAN: That's right. 01:12:49.550 --> 01:12:49.860 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:12:49.860 --> 01:12:50.355 align:middle line:90% BRYNMOR CHAPMAN: Yes. 01:12:50.355 --> 01:12:51.600 align:middle line:90% AUDIENCE: So i equals j. 01:12:51.600 --> 01:12:52.780 align:middle line:90% BRYNMOR CHAPMAN: Yes. 01:12:52.780 --> 01:12:56.240 align:middle line:90% So the answer was p is prime. 01:12:56.240 --> 01:12:59.050 align:middle line:90% So a and p are co-prime, right? 01:12:59.050 --> 01:13:00.950 align:middle line:90% So a has an inverse. 01:13:00.950 --> 01:13:03.250 align:middle line:90% We can cancel it. 01:13:03.250 --> 01:13:09.020 align:middle line:84% So this tells us that i is equivalent mod p to j. 01:13:09.020 --> 01:13:12.730 align:middle line:84% The two coefficients are equivalent. 01:13:12.730 --> 01:13:16.690 align:middle line:84% The two coefficients are coming from the set 1 to p minus 1. 01:13:16.690 --> 01:13:19.855 align:middle line:84% So if they're congruent mod p, they're actually equal. 01:13:19.855 --> 01:13:20.355 align:middle line:90% OK? 01:13:20.355 --> 01:13:26.290 align:middle line:90% 01:13:26.290 --> 01:13:29.170 align:middle line:84% So that means that all of these values 01:13:29.170 --> 01:13:32.320 align:middle line:84% are going to be distinct modulo p. 01:13:32.320 --> 01:13:33.290 align:middle line:90% None of them are 0. 01:13:33.290 --> 01:13:36.330 align:middle line:84% So they're these particular values. 01:13:36.330 --> 01:13:40.480 align:middle line:90% 01:13:40.480 --> 01:13:43.275 align:middle line:84% What happens when we multiply everything in S together? 01:13:43.275 --> 01:13:55.020 align:middle line:90% 01:13:55.020 --> 01:13:58.355 align:middle line:84% Well, if we multiply everything in this set, what do we get? 01:13:58.355 --> 01:14:08.490 align:middle line:90% 01:14:08.490 --> 01:14:09.480 align:middle line:90% Yeah? 01:14:09.480 --> 01:14:13.470 align:middle line:84% AUDIENCE: So is it p minus 1 and then the [INAUDIBLE]? 01:14:13.470 --> 01:14:15.790 align:middle line:84% BRYNMOR CHAPMAN: Yes, p minus 1 factorial, 01:14:15.790 --> 01:14:17.220 align:middle line:90% that's what it's called. 01:14:17.220 --> 01:14:21.370 align:middle line:84% So we take the product of both sides. 01:14:21.370 --> 01:14:23.550 align:middle line:90% We have the product of this-- 01:14:23.550 --> 01:14:29.610 align:middle line:84% or sorry, the product of this one 01:14:29.610 --> 01:14:32.976 align:middle line:84% is going to be equal to p minus 1 factorial. 01:14:32.976 --> 01:14:36.690 align:middle line:90% 01:14:36.690 --> 01:14:37.865 align:middle line:90% What about the other one? 01:14:37.865 --> 01:14:46.207 align:middle line:90% 01:14:46.207 --> 01:14:48.540 align:middle line:84% Well the other one is going to be the same, except we've 01:14:48.540 --> 01:14:50.740 align:middle line:90% got a bunch of factors of a. 01:14:50.740 --> 01:14:51.830 align:middle line:90% How many factors of a? 01:14:51.830 --> 01:14:54.820 align:middle line:90% 01:14:54.820 --> 01:14:55.712 align:middle line:90% Yeah? 01:14:55.712 --> 01:14:57.040 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:14:57.040 --> 01:14:57.957 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 01:14:57.957 --> 01:15:12.010 align:middle line:90% 01:15:12.010 --> 01:15:17.706 align:middle line:84% So this will be a to the p minus 1 times p minus 1 factorial. 01:15:17.706 --> 01:15:22.290 align:middle line:90% 01:15:22.290 --> 01:15:26.110 align:middle line:84% So p minus 1 factorial is equivalent to p 01:15:26.110 --> 01:15:29.500 align:middle line:84% minus 1 factorial times a to the p minus 1. 01:15:29.500 --> 01:15:30.540 align:middle line:90% What does that tell us? 01:15:30.540 --> 01:15:41.320 align:middle line:90% 01:15:41.320 --> 01:15:43.360 align:middle line:90% Yeah? 01:15:43.360 --> 01:15:47.120 align:middle line:84% AUDIENCE: Again, because p is prime, none of numbers from 1 01:15:47.120 --> 01:15:48.610 align:middle line:90% p minus 1 [INAUDIBLE] p. 01:15:48.610 --> 01:15:52.147 align:middle line:84% So p minus 1 factorial is [INAUDIBLE]. 01:15:52.147 --> 01:15:52.980 align:middle line:90% BRYNMOR CHAPMAN: OK. 01:15:52.980 --> 01:15:58.550 align:middle line:84% So the answer was that because p is prime, none of these numbers 01:15:58.550 --> 01:15:59.880 align:middle line:90% are a multiple of p. 01:15:59.880 --> 01:16:02.150 align:middle line:84% So their product is not a multiple of p. 01:16:02.150 --> 01:16:05.220 align:middle line:84% So p minus 1 factorial is not a multiple of p. 01:16:05.220 --> 01:16:07.910 align:middle line:90% So it has an inverse. 01:16:07.910 --> 01:16:11.570 align:middle line:84% So that means that we can cancel it from both sides. 01:16:11.570 --> 01:16:14.905 align:middle line:84% And we end up with 1 is congruent to a to the p minus 1. 01:16:14.905 --> 01:16:21.191 align:middle line:90% 01:16:21.191 --> 01:16:27.740 align:middle line:84% So we have p minus 1 factorial is congruent mod p 01:16:27.740 --> 01:16:33.320 align:middle line:84% to p minus 1 factorial times a to the p minus 1. 01:16:33.320 --> 01:16:34.075 align:middle line:90% These cancel. 01:16:34.075 --> 01:16:38.450 align:middle line:90% 01:16:38.450 --> 01:16:41.240 align:middle line:84% And we have 1 is congruent to the p minus 1. 01:16:41.240 --> 01:16:45.050 align:middle line:90% 01:16:45.050 --> 01:16:49.660 align:middle line:84% And that's the result that we were looking for. 01:16:49.660 --> 01:16:54.300 align:middle line:84% So if we're working mod n, we cannot generally reduce 01:16:54.300 --> 01:16:56.220 align:middle line:90% exponents mod n. 01:16:56.220 --> 01:17:00.545 align:middle line:84% But if n is prime, we can instead reduce mod n minus 1. 01:17:00.545 --> 01:17:05.010 align:middle line:90% 01:17:05.010 --> 01:17:06.670 align:middle line:84% Does that make sense to everybody? 01:17:06.670 --> 01:17:20.730 align:middle line:90% 01:17:20.730 --> 01:17:27.570 align:middle line:84% So as an example, so if we wanted 01:17:27.570 --> 01:17:36.210 align:middle line:84% to, say, find the remainder of 7 to the, 01:17:36.210 --> 01:17:50.205 align:middle line:84% say, 95 modulo, let's say, 13, how might we do that? 01:17:50.205 --> 01:17:59.680 align:middle line:90% 01:17:59.680 --> 01:18:02.920 align:middle line:90% Well, 13 is prime. 01:18:02.920 --> 01:18:05.920 align:middle line:84% So instead of computing 7 to the 95, 01:18:05.920 --> 01:18:11.830 align:middle line:84% let's reduce that mod 13 minus 1, so modulo 12. 01:18:11.830 --> 01:18:17.115 align:middle line:84% So this is going to be the same as 7 to the, what is it, 11? 01:18:17.115 --> 01:18:20.252 align:middle line:90% 01:18:20.252 --> 01:18:21.070 align:middle line:90% mod 13? 01:18:21.070 --> 01:18:28.353 align:middle line:90% 01:18:28.353 --> 01:18:29.145 align:middle line:90% Now what can we do? 01:18:29.145 --> 01:18:44.320 align:middle line:90% 01:18:44.320 --> 01:18:45.605 align:middle line:90% What is 7 to the 12? 01:18:45.605 --> 01:18:51.900 align:middle line:90% 01:18:51.900 --> 01:18:54.690 align:middle line:90% Sorry? 01:18:54.690 --> 01:18:58.140 align:middle line:90% Did somebody give an answer? 01:18:58.140 --> 01:18:59.272 align:middle line:90% No? 01:18:59.272 --> 01:19:00.690 align:middle line:90% AUDIENCE: It's 1. 01:19:00.690 --> 01:19:03.660 align:middle line:90% BRYNMOR CHAPMAN: 1, yes. 01:19:03.660 --> 01:19:08.370 align:middle line:90% So 7 to the 12 is 1. 01:19:08.370 --> 01:19:12.000 align:middle line:84% 7 to the 11 times 7 is 7 to the 12. 01:19:12.000 --> 01:19:16.685 align:middle line:84% So this is the modular inverse of 7 modulo 13. 01:19:16.685 --> 01:19:19.200 align:middle line:90% 01:19:19.200 --> 01:19:22.660 align:middle line:84% What can you multiply 7 by to get 1 mod 13? 01:19:22.660 --> 01:19:23.190 align:middle line:90% Yeah? 01:19:23.190 --> 01:19:23.820 align:middle line:90% AUDIENCE: 2. 01:19:23.820 --> 01:19:24.612 align:middle line:90% BRYNMOR CHAPMAN: 2. 01:19:24.612 --> 01:19:33.010 align:middle line:90% 01:19:33.010 --> 01:19:36.240 align:middle line:84% So that's all we have time for today. 01:19:36.240 --> 01:19:38.230 align:middle line:90% I will see you on Thursday. 01:19:38.230 --> 01:19:41.750 align:middle line:84% And if you have any questions, feel free to come up and ask. 01:19:41.750 --> 01:19:45.000 align:middle line:90%