<s>[INST] If $X$ and $Y$ are not divisible by 3, then $X^2 - Y^2$ is divisible by 3 Prove: If $X$ is not divisible by 3 and $Y$ is not divisible by 3, then $X^2 - Y^2$ is divisible by 3.
I cannot find a good way to prove this.. Tried contrapositive and contradiction but cannot see the answer. Anyone have any ideas?
 [/INST] If $x$ and $y$ are not divisible by 3 then they must be of the form $x = 3k \pm 1$ and $y = 3t \pm 1$. Therefore $x^2 - y^2 = 9k^2 \pm 6k + 1 -9t^2 \mp 6t -1 = 3(3k^2 - 3t^2 \pm 2k \mp 2t).$ So $x^2 - y^2$ is divisible by 3.
</s>