[Math] If $a$ and $b$ are odd then $a^2+b^2$ is not a perfect square

Question

Prove if $a$ and $b$ are odd then $a^2+b^2$ is not a perfect square.

We have been learning proof by contradiction and were told to use the Euclidean Algorithm.

I have tried it both as written and by contradiction and can't seem to get anywhere.

Answer 1

All squares are congruent to 1 or 0 mod 4. If they are odd they are congruent to 1 mod 4. therefore the sum of two odd squares is congruent to 2 mod 4. Thus not a square.