IMO 2011 Problem 5

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$ , the difference $f(m) - f(n)$ is divisible by $f(m- n)$ . Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$ , the number $f(n)$ is divisible by $f(m)$ .

Proposed by Mahyar Sefidgaran, Iran