Consider the problem of finding the greatest common divisor (gcd) of two positive integers...

Consider the problem of finding the greatest common divisor (gcd) of two positive integers a and b. The algorithm presented here is a variation of Euclid’s algorithm, which is based on the following theorem:4

THEOREM. If a and b are positive integers with a > b such that b is not a divisor of a, then gcd(a, b) = gcd(b, a mod b).

This relationship between gcd(a, b) and gcd( b, a mod b) is the heart of the recursive solution. It specifies how you can solve the problem of computing gcd(a, b) in terms of another problem of tire same type. Also, if b does divide a, then b = gcd(a, b?), so an appropriate choice for the base case is (a mod b) = 0. This theorem leads to the following recursive definition:

The following method implements this recursive algorithm:

a. Prove the theorem.

b. What happens if b > a?

c. How is tiie problem getting smaller? (That is, do you always approach a base case?) Why is the base case appropriate?