Exercise 2.5: Yet another definition of cons, car, cdr

Exercise 2.5. Show that we can represent pairs of nonnegative integers using only numbers and arithmetic operations if we represent the pair a and b as the integer that is the product 2^a3^b. Give the corresponding definitions of the procedures cons, car, and cdr.

This is possible because any number 2ⁿ will not contain 3 as a factor. Similarly, any number 3ⁿ will not contain 2 as a factor.

cons is a straightforward application of what the exercise describes.

(define (cons3 a b)
  (* (expt 2 a) (expt 3 b)))

car and cdr just repeatedly divide by 2 (car) or 3 (cdr) until the number no longer divides evenly. Count the number of even divisions, and return the count as the result.

(define (cdr3 c)
  (define (helper x count)
    (if (= 0 (remainder x 3))
        (helper (/ x 3) (+ 1 count))
        count))
  (helper c 0))

(define (car3 c)
  (define (helper x count)
    (if (= 0 (remainder x 2))
        (helper (/ x 2) (+ 1 count))
        count))
  (helper c 0))

;;tests
(car3 (cons3 23 67))
;Value: 23

(cdr3 (cons3 23 67))
;Value: 67