3.1. Among all integer solutions of the equation $20 x+19 y=2019$, find the one for which the value of $|x-y|$ is minimal. In the answer, write the product $x y$.
Answer: 2623.

Solution. One of the solutions to the equation is the pair $x=100, y=1$. Therefore, the set of all integer solutions is $x=100-19 n, y=1+20 n, n \in \mathbb{Z}$. The absolute difference $|x-y|=$ $|100-19 n-1-20 n|=|99-39 n|$ is minimized when $n=3$, and the corresponding solution is $(x, y)=(43,61)$. We write the answer as $x y=43 \cdot 61=2623$.