Question 28
Let $\mathbb{Z}_{\gt 0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b\gt C$,
$$
a+f(b) \mid a^{2}+b f(a)
$$
$$
a+f(b) \mid a^{2}+b f(a)
$$