MathNet 2018-part-03 — 181 problems (EN–VI) Question 28

Let $f:\{1,2,3, \ldots\} \rightarrow\{2,3, \ldots\}$ be a function such that $f(m+n) \mid f(m)+f(n)$ for all pairs $m, n$ of positive integers. Prove that there exists a positive integer $c\gt 1$ which divides all values of $f$.