RMM 2024 Problem 4

Fix integers $a$ and $b$ greater than 1 . For any positive integer $n$, let $r_{n}$ be the (non-negative) remainder that $b^{n}$ leaves upon division by $a^{n}$. Assume there exists a positive integer $N$ such that $r_{n}<2^{n} / n$ for all integers $n \geq N$. Prove that $a$ divides $b$.