Question 54
$k \in \mathbb{Z}^+$ is a fixed number. Find all functions $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that for infinitely many prime numbers like $q$, $q^k$ is in the range of $f$ and also for all $m, n \in \mathbb{Z}^+$
$$
f(m) + f(n) \mid f(m + n)
$$
$$
f(m) + f(n) \mid f(m + n)
$$