BALKAN 2012 Problem 4

$Let$\mathbb{Z}^{+}$be the set of positive integers. Find all functions$f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$such that the following conditions both hold: (i)$f(n!)=f(n)$! for every positive integer$n$, (ii)$m-n$divides$f(m)-f(n)$whenever$m$and$n$ are different positive integers.