MHB Sufficient Proof: $4 \nmid n^2 - 2$ for $r=0$

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The discussion centers on proving that for integers n, the expression $4 \nmid n^2 - 2$ holds true when r = 0 in the context of the quotient remainder theorem. It is established that if n is a multiple of 4, then $n^2$ results in 0 mod 4, leading to $n^2 - 2$ being congruent to 2 mod 4, which is not divisible by 4. The argument is further supported by the observation that all perfect squares are either 0 or 1 mod 4, confirming that $n^2 - 2$ can only yield results of 2 or 3 mod 4. Thus, the conclusion is reached that for the case of r = 0, it is indeed sufficient to state that $4 \nmid n^2 - 2$.
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I would like to prove that this is incorrect:

$\exists x \in \Bbb{Z}$ such that $ 4 | n^2 - 2$

I can use the quotient remainder theorem,

$n = dq + r$ where $ 0 <= r < d $ and $ d = 4$

For the case $ r = 0$ is this sufficient proof?

$n = 4q $ and $4 | n^2 - 2$ thus $4 | 16q^2 - 2$

then $4 | 4(4q^2) + 2$

which can't be true, $\therefore $ for the case $ r = 0$, $4 \nmid n^2 - 2$

Is this sufficient for the case $r = 0$? (I can figure out the rest of the cases from here)
 
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There's no need to use the "quotient remainder theorem" (unless you've been directed to do so).

It's easily proved that all perfect squares are either 0 or 1 mod 4, thus n2 - 2 is 2 or 3 mod 4.
 
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