How many pairs of integers $x, y$, between 1 and 1000, exist such that $x^{2}+y^{2}$ is divisible by 7.
The number $x^{2}+y^{2}$ is divisible by 7 if and only if both numbers $x$ and $y$ are multiples of 7 (see problem 108745). The number of integers between 1 and 1000 that are multiples of 7 is 142 (since $1000=142 \cdot 7+6$). Therefore, the desired number is $142^{2}$.

## Answer

$142^{2}=20164$ pairs.