Proving the Impossibility of Odd Pythagorean Triples

[saadsarfraz]

If we have a pythagorean triple a^2 + b^2 = c^2 and we need to show that a and b both cannot be odd. I found a proof from a website:

if a and b both odd, then we must have c$$^{2}$$$$\equiv$$a$$^{2}$$+b$$^{2}$$$$\equiv$$1+1$$\equiv$$2 (mod4), which is a contradiction, since 2 is not a square mod 4. Hence at least one of a and b must be even.

I didnt quite understand the proof as this is just when a and b are 1? what about other odd numbers.

 

[Focus]

If a is odd then $$a^2=[1]$$ in $$\mathbb{Z}_4$$ (The only odds in there are 1 and 3, both give 1 when squared).