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