USAMO 2018 Problem 3

For a given integer $n\ge 2$ , let $\{a_1,a_2,…,a_m\}$ be the set of positive integers less than $n$ that are relatively prime to $n$ . Prove that if every prime that divides $m$ also divides $n$ , then $a_1^k+a_2^k + \dots + a_m^k$ is divisible by $m$ for every positive integer $k$ .

Proposed by Ivan Borsenco