Question about parity of Pythagorean triples

Question

I want to show that if $(a,b,c)\in\mathbb{N}^+\times \mathbb{N}^+\times \mathbb{N}^+$ is a Pythagorean triple with $gcd(a,b,c)=1$ then $a$ and $b$ cannot simultaniously be odd.

Reducing modulo $4$ gives an easy and quick way to show this but I would like to show it without modulo reduction (I mean this exercise is in Beachy and Blair: Abstract algebra, and the whole congruence class story is in the next chapter)

Any HINTS for me?

Thank you in advance

Answer 1

You can assume $a=2n+1$ and $b=2m+1$ and use $a^2+b^2=c^2$ taking two cases, one assuming $c=2p$ and another assuming that $c=2p+1$. Contradictions will arise.