For positive real numbers $p$ and $q$, define the remainder when dividing $p$ 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 remainders 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 $r_1 + r_2 = \sqrt{2}$.

Let's think step by step and output the final answer within \boxed{}.
user