10.2. How many solutions in natural numbers $x, y$ does the equation $x+y+2 x y=2023$ have?
Answer: 6.

Solution. Multiply the equation by 2, add one to both sides, and factor the left side, while expressing the right side as a product of prime factors:

$$
2 x+4 x y+2 y+1=4047 \Leftrightarrow(2 x+1)(2 y+1)=4047 \Leftrightarrow(2 x+1)(2 y+1)=3 \cdot 19 \cdot 71
$$

Since \(x\) and \(y\) are natural numbers, each factor on the left side is at least three, so the number one cannot appear in the factorization options, and the possible options are: