The number of solutions of $x^{22} \equiv 2(mod23)$ has-

Question

Possible answers-

a).no solutions.

b).$23$ solutions.

c).exactly one solution.

d).$22$ solutions.

Solution: Since,$gcd(1,23)=23$ divides 2,so it has exactly one solution.

It seems i'm wrong because the concept which i've applied is for linear congrueces.

Don't know how to pursue this.

Is there any theorem generalizing this result?

Need help!

Answer 1

Hint $\ x\equiv 0\,$ is not a root, nor is any $\,x\not\equiv 0\,$ a root, since then $\,x^{22}\equiv 1\,$ by little Fermat

Answer 2

If $(x,23)=1$ using Fermat's Little Theorem, $$x^{23-1}\equiv1\pmod{23}$$

and $$1\not\equiv2\pmod{23}$$

What if $(x,23)\ne1\iff23|x$?