<s>[INST] Does $7$ divide $2 x^2 - 4y^2$ for all $x,y$? Does $7$ divide $2 x^2 - 4y^2$ for all $x,y \in \mathbb{Z}$?
 [/INST] ...suppose the result holds true, then, 
$2x^2-4y^2=0 \pmod7$ 
since $2$ does not divide $7$, we have $x^2-2y^2=0 \pmod7$, but $2$ is congruent to $9$ modulo $7$, so, $x^2-9y^2=0 \pmod7$ or 
$x=3y \pmod7$, so the result holds true $iff$ $x=3y+7k$, for some integer $k$
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