Problem

Source: XII International Festival of Young Mathematicians Sozopol 2023, Theme for 10-12 grade

Tags: number theory



Let $n \geq 2$ be an integer such that $6^n + 11^n$ is divisible by $n$. Prove that $n^{100} + 6^n + 11^n$ is divisible by $17n$ and not divisible by $289n$.