How to solve $x^2+y^2 \equiv 8\pmod 9$?

Question

How to solve $x^2+y^2 \equiv 8 \pmod 9$?

I know the Chinese Remainder Theorem, but how do I apply it here?

Answer 1

If we consider each possible residue of $x^2\mod9$, we see that they are

$0^2\equiv 0$

$1^2\equiv 1$

$2^2\equiv 4$

$3^2\equiv 0$

$4^2\equiv 7$

$5^2\equiv 7$

$6^2\equiv 0$

$7^2\equiv 4$

$8^2\equiv 1$

Obviously (in a minor abuse of notation) the solutions have to be $x\in\{4,5\},y\in\{1,8\}$ (or vice versa) (where numbers represent their residue classes) or $x,y\in\{2,7\}$.