We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b(modn)$. We investigate the structure of $G(n,k)$. In particular, upper bounds are given for the longest cycle in $G(n,k)$. We find subdigraphs of $G(n,k)$, called fundamental constituents of $G(n,k)$, for which all trees attached to cycle vertices are isomorphic.

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(mod2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product ${\prod }_{k\in R_m\left(p\right)}(1+tan(\pi ak/p))$, where $$R_m\left(p\right)=\{0<k<p:k\in \mathbb{Z}\text{is}\text{an}m\text{th}\text{power}\text{residue}\text{modulo}p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4\left(p\right)}\left(1+tan\pi \frac{ak}{p}\right)={(-1)}^y{(-2)}^{(p-1)/8}.$$