Question 28
Let $\mathbb{Z}_{\gt 0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ is called $k$-good if $\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.