Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.
Source: III Caucasus Mathematical Olympiad
Tags: number theory
Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.