If $x^2\equiv1\pmod5$, what can be said about $x \pmod5$?

Question

(ENC 2000) If $x^2\equiv1\pmod5$, $x\in\mathbb{N},$ then:

A) $x\equiv1\pmod5$

B) $x\equiv2\pmod5$

C) $x\equiv4\pmod5$

D) $x\equiv1\pmod5$ or $x\equiv4\pmod5$

E) $x\equiv2\pmod5$ or $x\equiv4\pmod5$

I tried $$x^2\equiv1\pmod5\Longrightarrow5\mid1-x^2\Longrightarrow5\mid(1+x)(1-x)$$ Hence $$5\mid(1-x)\;\;\text{or}\;\;5\mid(1+x)$$ If $$5\mid1-x\Longrightarrow x\equiv1\pmod5$$ If $$5\mid 1+x\;\;?$$

Answer 1

$$x^2\equiv1\pmod5\Longrightarrow5\mid1-x^2\Longrightarrow5\mid(1+x)(1-x)$$$$\text{Soon}\;\;\;5\mid(1-x)\;\;\text{or}\;\;5\mid(1+x)$$$$\text{If}\;\;\;5\mid1-x\Longrightarrow x\equiv1\pmod5$$ $$\text{If}\;\;5\mid 1+x\Longrightarrow x\equiv -1\pmod5\Longrightarrow x\equiv4\pmod5$$$$$$Alternative (D)

Answer 2

Hint:

You know $x \equiv 1 \pmod 5$ is one possibility. What is $4^{2}\pmod 5$? Are there any others possibilities such that $x^{2} \equiv 1 \pmod 5$ in the list?

Answer 3

Hints:

$$x^2=1\iff x^2-1=0\iff (x-1)(x+1)=0\iff x=\pm 1$$

The above is true in any field...