US-EGMO-TST 2019 Problem 3

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.