KURSCHAK 2011 Problem 1

Let $a_1, a_2,...$ be an infinite sequence of positive integers such that for any $k,\ell\in \mathbb{Z_+}$ , $a_{k+\ell}$ is divisible by $\gcd(a_k,a_\ell)$ . Prove that for any integers $1\leqslant k\leqslant n$ , $a_na_{n-1}\dots a_{n-k+1}$ is divisible by $a_ka_{k-1}\dots a_1$ .