Question 21
Let $\mathbb{Z}_{\gt 0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers $m$ and $n$.
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers $m$ and $n$.