TSTST 2022 Problem 4

A function $f: \mathbb{N} \rightarrow \mathbb{N}$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers $$f(m+1), f(m+2), \ldots, f(m+f(n))$$ is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$.