Show that if $x^2 ≡ 0 \pmod p$ where $p$ is prime then $p \mid x$.
$x^2 \equiv 0 \pmod p$ means that $p \mid x \times x$.
Let's recall Proposition (2.2), which says that if $p$ is prime and $p \mid (a × b)$ then $p \mid a$ or $p \mid b$.
Since $p$ is prime, $p \mid x \times x$ means $p \mid x$.
Alternatively, we can use Proposition (3.14) that if $a × b ≡ 0 \pmod p$ where $p$ is prime then $a ≡ 0 \pmod p$ or $b ≡ 0 \pmod p$.
So $x \times x \equiv 0 \pmod p$ means $x = 0 \pmod p$, which means $p \mid x$.