elementary number theory - $ 7mid x text{ and } 7mid y Longleftrightarrow 7mid x^2+y^2 $

Show that
$$ 7\mid x \text{ and } 7\mid y \Longleftrightarrow 7\mid x^2+y^2 $$

Indeed,

First let's show

$7\mid x \text{ and } 7\mid y \Longrightarrow 7\mid x^2+y^2 $

we've $7\mid x \implies 7\mid x^2$ the same for $7\mid y \implies 7\mid y^2$ then
$ 7\mid x^2+y^2 $

• Am i right and can we write $a\mid x \implies a\mid x^P ,\ \forall p\in \mathbb{N}^*$

Now let's show

$7\mid x^2+y^2 \Longrightarrow 7\mid x \text{ and } 7\mid y$

$7\mid x^2+y^2 \Longleftrightarrow x^2+y^2=0 \pmod 7 $

for

\begin{array}{|c|c|c|c|c|} \hline
x& 0 & 1 & 2& 3 & 4 & 5 & 6 \\ \hline
x^2& 0 & 1 & 4& 2 & 2 & 4 & 1 &\pmod 7\\ \hline
y& 0 & 1 & 2& 3 & 4 & 5 & 6 \\ \hline

y^2& 0 & 1 & 4& 2 & 2 & 4 & 1 & \pmod 7 \\ \hline
\end{array}

which means we have one possibility that $x=y= 0 \pmod 7 $

• Am I right and are there other ways?