Primitive Pythagorean triple divisible by 3 Prove that for any primitive Pythagorean triple (a, b, c), exactly one of a and b must be a multiple
of 3, and c cannot be a multiple of 3.
My attempt:
Let a and b be relatively prime positive integers.
If $a\equiv \pm1 \pmod{3}$ and $b\equiv \pm1 \pmod{3}$, 
$c^2=a^2+b^2\equiv 1+1\equiv 2 \pmod{3}$
This is impossible as the only quadratic residues modulo 3 are 0 and 1.
So far, so good.
If one of a, b is $\equiv 0 \pmod{3}$ and the other is $\equiv \pm1 \pmod{3}$,
$c^2=a^2+b^2\equiv 0+1\equiv 1 \pmod{3}$
This is the part I don't understand. Just because $c^2\equiv 1\pmod{3}$ doesn't mean that $c^2$ must be a perfect square. For example, $a=12$ and $b=13$ satisfy the above conditions but $c^2=a^2+b^2=313$, which isn't a perfect square.

Your proof is complete. You are asked to show that one of $a,b,c$ is divisible by $3$. In the first part you show that $a$ and $b$ can't both be non-divisible by $3$. In the second part, you assume that one of $a,b$ is divisible by $3$ and show that $c^2\equiv 1$ (mod $3$) which implies that $c$ is not divisible by $3$ and hence exactly one of $a,b,c$ is divisible by $3$.
