<s>[INST] Suppose $n$ is prime and $x \in Z$ satisfies $x^2 \equiv 1 \mod n.$ Prove that $x \equiv 1 \mod n$ or $x \equiv -1 \mod n$. Suppose $n$ is prime and $x \in Z$ satisfies $x^2 \equiv 1 \mod n.$ Prove that $x \equiv 1 \mod n$ or $x \equiv -1 \mod n$. 
So far I have done the following proof, but I am unsure how to complete it: 
We will begin by explaining why $x^2 \equiv 1 \mod n$ iff $n|(x^2 - 1)$. Then use this to prove that $x \equiv 1 \mod n$ or $x \equiv -1 \mod n$.
If $x^2 \equiv 1 \mbox{ mod } n$, then $x^2 = nk +1 $, for some $k \in Z$. Therefore $nk = x^2 -1$, and $n|(x^2 -1)$. 
On the other hand, if $n|(x^2 -1)$, then $nk = x^2 -1$ for some $k \in Z$, and then we can say $x^2 = nk +1$, and thus $x^2 \equiv 1 \mod n$. 
From here, however I am not sure how to demonstrate that this shows $x \equiv 1 \mod n$ or $x \equiv -1 \mod n$. 
 [/INST] You can factor $(x^2-1)=(x+1)(x-1)$.  Since $n$ is prime, $n\mid (x^2-1)$ implies $n\mid x+1$ or $n\mid x-1$.  So $x\equiv 1$ mod $n$ or $x\equiv -1$ mod $n$.
</s>