Show that if $\langle x_0\rangle$ is a $n$-cycle and $n$ is a prime number, then $\langle x_0\rangle$ is a prime $n$-cycle.
Recall: a natural number $n\neq 1$ is a prime number if there are no proper divisors of $n$; the latter means if $m\in\mathbb{N}$ and $\frac{n}{m}$, then $m=1$ or $m=n$.
(Hint: given that $\langle x_0\rangle$ is a $n$-cycle, let $m$ be the first natural number larger than $1$ with $x_m=x_0$. Show that $m$ divides $n$).
I know to be a $n$-cycle it must satisfy $\langle x_0\rangle = ${$x_0,x_1,...,x_{n-1}$} with $x_0=F(x_{n-1})$
and to be a prime $n$-cycle it must also satisfy $x_i \neq x_0$ for $i=1,2,...,n-1$
How does one prove the above statements with that knowledge?