1. The problem statement, all variables and given/known data Theorem: Let A, B, C, M, and N be integers. If A divides both B and C, A divides NB+MC. 2. Relevant equations 3. 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.