Problem Link:

Practice
Contest

Difficulty:

CAKEWALK

Pre-requisites:

gcd

Problem:

You are given a set of N positive numbers A[0], A[1], ..., A[N-1]. At each step you pick two unequal numbers and replace the larger one by the difference. You stop when all numbers are equal. Find at what number will you stop.

Explanation:

You stop at the GCD of the numbers. To prove this, we just use the fact that gcd(a, b) = gcd(a-b, b), where a > b. Now, let the original GCD be G. After a single step, since gcd(a, b) = gcd(a-b, b), in particular, gcd(A[i], A[j]) = gcd(A[i]-A[j], A[j]), we get that the gcd remains G.

If we finally end at the number K, then gcd(K, K, K, ..., K) = K. But since after each step the gcd remains invariant, we get that K = G.

Setter's Solution:

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Tester's Solution:

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