Prove $7\mid x^2+y^2$ iff $7\mid x$ and $7\mid y$ The question is basically in the title: Prove $7\mid x^2+y^2$ iff $7\mid x$ and $7\mid y$
I get how to do it from $7\mid $ and $7\mid y$ to $7\mid x^2+y^2$, but not the other way around.
Help is appreciated! Thanks.

$x^2,y^2$  can be $0^2\equiv0, (\pm1)^2\equiv1,(\pm2)^2\equiv4, (\pm3)^2\equiv2\pmod 7$
Observe that for no combination except $0,0$ of $x^2+y^2 \equiv0\pmod 7$

Alternatively, 
If $(7,xy)=1, x^2+y^2\equiv0\pmod 7\implies \left(\frac xy\right)^2\equiv-1\pmod 7$
But we  know $-1$ is a Quadratic residue $\pmod p$ iff prime $p\equiv 1\pmod 4$
