Here is my question:
Let $\zeta$ be the first $n$-th root of unity, and let $\eta$ = $\zeta^k$ be any other $n$-th root of unity. Show that there exists an integer $m$ such that $\zeta = \eta^m$ if and only if $k$ has a multiplicative inverse in $\mathbb{Z}_n$.
Today was my first lecture in working with modular addition, subtraction, and multiplication. I know how to compute equation in $\mathbb{Z}_5$ for example (previous homework exercise based of solving an equation in $\mathbb{Z}_5$ but when I came across this proof, I got stuck. I don't even know how to begin the proof since this isn't about solving equations. It's one of those exercises that are one of the last ones before the end of a section and I even asked my friend if he knew where to start but after $30$ minutes, we had no success.