MathNet 2012 Question 99

Let $\mathbb{Z}^+$ be the set of all positive integers. Find all functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $f(n!) = f(n)!$, and $m-n$ divides $f(m)-f(n)$ for all distinct positive integers $m$ and $n$.