51st Ukrainian National Mathematical Olympiad, 3rd Round

Question

Suppose that for two natural numbers m, n the following equality holds m+n = [m,n] + (m,n), Where [m,n] and (m,n) are the least common multiple and the greatest common divisor of m,n respectively. Prove that one number is divisible by another.

Accepted Answer

Let d = (m,n), then m = ad, n = bd, and using the formula mn = [m,n] (m,n) we get, that [m,n] = abd. This implies that abd + d = ad + bd d(a-1)(b-1) = 0, which is possible if either a=1 or b=1 and the result follows.