Given $x^{93} \equiv 2 \, (\textrm{mod}\ 5)$, $x \equiv \,? \, (\textrm{mod}\ 5)$

Question

Given $x^{93} \equiv 2 \, (\textrm{mod}\ 5)$

$x \equiv \,? \, (\textrm{mod}\ 5)$

Answer 1

Without loss of generality, let $\gcd(x,5) = 1$. From Euler's Theorem we have $x^4 \equiv 1 (\mod 5)$. Hence

$$ x^{93} \equiv (x^{4})^{23}.x \equiv x \equiv 2 (\mod 5) $$

Answer 2

Hint:

You just have to try $5$ numbers.

Try $x$ being $0,1,2,-1,-2$.

Answer 3

Hint Notice that $2^4\equiv 1[5]$ and $93=4\times 23+1$. Can you take it from here?

Answer 4

Hint: If $x \not\equiv 0 \mod 5$, $x^{4n} \equiv 1 \mod 5$ for any integer $n$.