Nemotron-IMO-Bench 2026 N50

A function $f:\mathbf{N}\to \mathbf{N}$ has the property that $\gcd(f(m), f(n))=1$ whenever $\gcd(m,n)=1$, and $n\leq f(n)\leq n+2012$ for all $n$. Prove that if $n>1$ then any prime divisor of $f(n)$ is a prime divisor of $n$.