BALTIC-WAY 2020 Problem 20

Let $A$ and $B$ be sets of positive integers with $|A| \geq 2$ and $|B| \geq 2$. Let $S$ be a set consisting of $|A|+|B|-1$ numbers of the form $a b$ where $a \in A$ and $b \in B$. Prove that there exist pairwise distinct $x, y, z \in S$ such that $x$ is a divisor of $y z$.