Question 163
Let $\mathbb{Z}_{ \gt 0}$ denote the set of positive integers. Let $f: \mathbb{Z}_{ \gt 0} \rightarrow \mathbb{Z}_{ \gt 0}$ be a function satisfying the following property: for $m, n \in \mathbb{Z}_{ \gt 0}$, the equation
$$
f(m n)^2 = f\left(m^2\right) f(f(n)) f(m f(n))
$$
holds if and only if $m$ and $n$ are coprime.
For each positive integer $n$, determine all the possible values of $f(n)$.
$$
f(m n)^2 = f\left(m^2\right) f(f(n)) f(m f(n))
$$
holds if and only if $m$ and $n$ are coprime.
For each positive integer $n$, determine all the possible values of $f(n)$.