Question 172
Find all functions $f : \mathbb{N}^* \to \mathbb{N}^*$ fulfilling the relation
$$
d(x, f(y)) \cdot m(f(x), y) = d(x, y) \cdot m(f(x), f(y)), \text{ for every } x, y \in \mathbb{N}^*,
$$
where $d(a, b)$ and $m(a, b)$ signify the greatest common divisor, respectively the lowest common multiple of the positive integers $a$ and $b$.
$$
d(x, f(y)) \cdot m(f(x), y) = d(x, y) \cdot m(f(x), f(y)), \text{ for every } x, y \in \mathbb{N}^*,
$$
where $d(a, b)$ and $m(a, b)$ signify the greatest common divisor, respectively the lowest common multiple of the positive integers $a$ and $b$.