CN-TST 2017 Problem 7

Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$ . Show that there exists a natural $N$ such that for all $n \geq N$ , one has $f(n) \leq n^{0.01}$ .