Problem

Source: USA TST for EGMO 2019, Problem 3 (adapted from IMO TST Problem 2)

Tags: number theory



Let $n$ be a positive integer such that the number \[\frac{1^k + 2^k + \dots + n^k}{n}\]is an integer for any $k \in \{1, 2, \dots, 99\}$. Prove that $n$ has no divisors between 2 and 100, inclusive.