IN-TST 2001 Problem 14

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$ . Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$ . Prove that $r | k$ and $k | s$ .