6.090 IAP '05 - Homework 3 Solutions

Problem 1

1.
(define with-tax
  (lambda (x)
    (* x 1.05)))

2.
(define num-digits
  (lambda (n)
    (if (< n 10)
        1
        (+ 1 (num-digits (quotient n 10))))))

3.
(define add-digit
  (lambda (n digit)
    (+ (* n 10) digit)))

Problem 2 (Reasonable)

1.
(define divisible?
  (lambda (x y)
    (= (remainder x y) 0)))

2.
(define slow-add
  (lambda (a b)
    (if (= a 0)
        b
        (inc (slow-add (dec a) b)))))
or
(define slow-add
  (lambda (a b)
    (if (= a 0)
        b
        (slow-add (dec a) (inc b)))))

Problem 3 (Sticky)

1.
(define smallest-factor
  (lambda (n f)
    (if (= n f)
        #f       ; tested all possible factors!
        (if (divisible? n f)
            f    ; found a factor!
            (smallest-factor n (+ f 1))))))

2.
(define prime?
  (lambda (n)
    (if (smallest-factor n 2)
        #f
        #t)))