If an integer divides the greadest common divisor of two integers, then it divides any linear combination of the two integers

Let a,b,k be integers. Prove that if k divides a and b, then k divides as + bt for all integers s and t.

Suppose k divides a and b; then there exist integers n and m such that a = kn and b = km. Now we have as + bt = kns + kmt = k(ns + mt) for all s,t, so k divides as + bt.

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