Problem

Source: VI Caucasus Mathematical Olympiad

Tags: number theory



Let $a, b, c$ be positive integers such that the product $$\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a) $$is a perfect square. Prove that the product $$\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a) $$is also a perfect square.