In Section 1.2.3, we studied Euclid’s algorithm for computing the greatest common divisor...

In Section 1.2.3, we studied Euclid’s algorithm for computing the greatest common divisor (gcd) of two positive integers: the largest integer which divides them both. Here we will look at an alternative algorithm based on divide-and- conquer.

(a) Show that the following rule is true.

(b) Give an efficient divide-and-conquer algorithm for greatest common divisor.

(c) How does the efficiency of your algorithm compare to Euclid’s algorithm if a and b are n-bit integers? (In particular, since n might be large you cannot assume that basic arithmetic operations like addition take constant time.)