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## Homework Statement

Theorem:

Let A, B, C, M, and N be integers.

If A divides both B and C, A divides NB+MC.

## Homework Equations

## The Attempt at a Solution

Proof:

Since we have defined A to divide both B and C, there exists an M in the integers such that B = AM, and there exists an N in the integers such that C = AN.

So, it follows that:

A = B/M and A = C/N

B/M + C/N = A + A

B/M + C/N = 2A

B + MC/N = 2AM

NB + MC = 2AMN

It is shown that 2AMN = NB + MC, and since A divides 2AMN, A divides NB+MC.

Therefore, A divides NB + MC.