Primitive roots as roots of equations. 
Take $g$ to be a primitive root $\pmod p$, and $n \in \{0, 1,\ldots,p-2\}$ write down a necessary sufficient condition for $x=g^n$ to be a root of $x^5\equiv 1\pmod p$ . This should depend on $n$ and $p$ only, not $g$.
How many such roots $x$ of this equation are there?  This answer may only depend on $p$.

At a guess for the first part I'd say as $g^{5n} \equiv g^{p-1}$ it implies for $x$ to be a root $5n \equiv p-1 \pmod p$.  No idea if this is right and not sure what to do for second part.  Thanks for any help.

$g^{m} \equiv 1 \mod p$ if and only if $\mathrm{ord}_p(g)$ divides $m$. 
Since $g$ is primitive root, we get that $p-1=\mathrm{ord}_p(g)$ has to divide $5n$.
Can you finish the problem now? 
