Question 188
Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers with the following property: if $S$ is any nonempty set of positive integers, there exists $s \in S$ with $a_{s} \leq \operatorname{gcd}(S)$. Prove that $n!$ is divisible by $a_{1} a_{2} \cdots a_{n}$ for every positive integer $n$.