if $(a,b,c)$ is a Pythagorean triple then $a$ or $b$ or $c$ can be divided by 3 Prove that if $(a,b,c)$ is a Pythagorean triple then $a$ or $b$ or $c$ can be divided by 3

We know that $a^2+b^2=c^2$, and that the perfect squares mod $3$ are $0$ and $1$. Therefore $c^2$ is not $2$ mod $3$, and so it's not the case that both $a^2$ and $b^2$ are $1$ mod $3$. Therefore one of them is $0$ mod $3$. Let it be $a^2$. Then $a$ is also $0$ mod $3$ because $1^2=2^2=1\pmod{3}$. So $3|a$.
