Expressing Irrationality of Square Root

Question

I fully understand the proof. However, at this stage:

$a^2 = 3b^2$

I understand that both cannot be even since they don't have common factors, but why does that lead to them both being odd? What it one was odd and one was even? Is there a proof that is not like that? Thank you.

Answer 1

If one of them is odd, the other is too:

• If $a$ is odd, $a^2$ is too, i.e. $3b^2$ is odd, which implies $b$ is odd.
• If $b$ is odd, $3b^2=a^2$ is, hence $a$ is odd.

In either case, both $a$ and $b$ are odd.