Question 31
Let $\mathbb{Z}_{ \gt 0}$ denote the set of positive integers. Let $f: \mathbb{Z}_{ \gt 0} \to \mathbb{Z}_{ \gt 0}$ be a function satisfying the following property: for $m, n \in \mathbb{Z}_{ \gt 0}$, the equation
$$ f(mn)^{2}=f(m^{2})f(f(n))f(mf(n)) $$
holds if and only if m and n are coprime.
For each positive integer n, determine all the possible values of $ f(n) $. (Japan)
$$ f(mn)^{2}=f(m^{2})f(f(n))f(mf(n)) $$
holds if and only if m and n are coprime.
For each positive integer n, determine all the possible values of $ f(n) $. (Japan)