Prove that if $a \mid b$ and $a \mid c$ then $a \mid (b^2− c^2)$.
We're given
$$b = am$$
$$c = an$$
for some integers $m,n$.
We then have
$$ b^2 - c^2 = a(am^2 - an^2)$$
which means $a \mid (b^2− c^2)$.