Let $p$ be prime, $k ∈ \mathbb{N}$ and suppose that $\gcd(k,p-1)=d$. Show that $x^k ≡ 1\pmod{p}$ has $d$ distinct solutions modulo $p$.

Question

I'm lost here. Any help would be great.

Let $p$ be prime, $k ∈ \mathbb{N}$ and suppose that $\gcd(k,p-1)=d$. Show that $x^k ≡ 1\pmod{p}$ has $d$ distinct solutions modulo $p$.

Answer 1

If you know that there is a primitive root mod $p$, then this is easy: the solutions are $g^{j\frac{p-1}{d}}$ for $j=0,\dots,d-1$, where $g$ is a primitive root.

Start by proving that $x^k ≡ 1\bmod{p} \iff x^d ≡ 1\bmod{p}$.