<s>[INST] Proving that if $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even. 
Let $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even.

My attempt:
If one or two numbers of $a,b,c$ are even then we're done, so we'll have to show that at least one of them is even. 
Suppose $a,b$ are odd, then $a^2,b^2$ are odd, so $a^2+b^2$ must be even. So $c^2$ is even then $c$ is even. So $abc$ is even.
Is this enough to prove it?
 [/INST] Notice that a square of an integer is either $0$ or $1$ modulo $4$ therefore. If $$c^2\equiv0\mod{4}\Rightarrow c\equiv0\mod{2}\Rightarrow abc\equiv0\mod{2}$$
If $$c^2\equiv1\mod{4}\Rightarrow a^2\equiv0, b^2\equiv1\mod{4}\Rightarrow a\equiv0\mod{2}\Rightarrow abc\equiv0\mod{2}$$
or
$$a^2\equiv1, b^2\equiv0\mod{4}\Rightarrow b\equiv0\mod{2}\Rightarrow abc\equiv0\mod{2}$$
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