Question 97
Find all functions $f : \mathbb{N} \setminus \{1\} \rightarrow \mathbb{N}$ such that for all distinct $x, y \in \mathbb{N}$ with $y \ge 2018$,
$$
\gcd(f(x), y) \cdot \operatorname{lcm}(x, f(y)) = f(x)f(y)
$$
$$
\gcd(f(x), y) \cdot \operatorname{lcm}(x, f(y)) = f(x)f(y)
$$