HMMT 2019 A6

For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_{1}+r_{2}=\sqrt{2}$.