Number of solutions to $x^k ≡ n (mod\ p)$

Question

I am asked to show: If $(n,p)=1$ show that the equation $x^k\equiv{n}\mod p$ has either no solutions or exactly k solutions.

How do I approach this problem?

I've begun by defining the obvious congruence, i.e.: this means $p | x^k-n$ but wasn't sure how to proceed

Answer 1

Part a) is not true. You have infinitely many solutions, unless you restrict to $x<p$. And in this last case, for instance, $x^3\equiv 1\pmod 3$ has one solution, since $1^3=1$ but $2^3\equiv 2\pmod 3$. For a more dramatic example, $x^5\equiv 1\pmod 7$ has only $1$ as solution.

And the case $n=1$ shows that b) is not true either.