Question 34
Find all functions $f: \mathbb{Z}_{\gt 0} \to \mathbb{Z}_{\gt 0}$ such that, for all positive integers $m$, $n$, and $a$, the following holds:
1. $f(f(m)f(n)) = mn$;
2. $f(2024a + 1) = 2024a + 1$.
1. $f(f(m)f(n)) = mn$;
2. $f(2024a + 1) = 2024a + 1$.