The irrationality of $\sqrt{(}2)$

Consider a rational number fully reduced, $\frac{p}{q}$ such that $\frac{p^2}{q^2}=2$. Investigate whether $p$ or $q$ are odd or even numbers leads to a contradiction which shows that $2$ cannot be rational.

• $p$ and $q$ can't both be even, as [...].
• $p$ and $q$ must both be odd, as [...].
• $p$ is even as [...].

These points lead to a contradiction.