Question 81
Fix integers $a$ and $b$ greater than $1$. For any positive integer $n$, let $r_n$ be the (non-negative) remainder $b^n$ leaves upon division by $a^n$. Assume that there exists a positive integer $N$ such that $r_n \lt 2^n/n$ for all integers $n \ge N$. Prove that $a$ divides $b$.