Question 152
Suppose that we have three natural numbers $a$, $b$ and $c$ such that one of the values
$$
\gcd(a, b) \cdot \text{lcm}(b, c), \gcd(b, c) \cdot \text{lcm}(c, a), \gcd(c, a) \cdot \text{lcm}(a, b),
$$
is equal to the product of the other two. Prove that one of the numbers $a$, $b$ and $c$ is a multiple of a different one.
$$
\gcd(a, b) \cdot \text{lcm}(b, c), \gcd(b, c) \cdot \text{lcm}(c, a), \gcd(c, a) \cdot \text{lcm}(a, b),
$$
is equal to the product of the other two. Prove that one of the numbers $a$, $b$ and $c$ is a multiple of a different one.