<s>[INST] How to solve $x^2+y^2 \equiv 8\pmod 9$? How to solve $x^2+y^2 \equiv 8 \pmod 9$?
I know the Chinese Remainder Theorem, but how do I apply it here?
 [/INST] 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 $x,y\in\{2,7\}$.
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