Prove that function $g(n)=\zeta^{5n}$ is surjective from $\mathbb Z$ to the set of roots of unity

Question

I have this math problem, that I'm kind of stuck on.

$\mu_{102} = \{ z \in \mathbb{C}: z^{102} = 1\}$

Let $\zeta = e^{\frac{2 \pi i}{102}}.$ Define $g : \mathbb{Z} \to \mu_{102}$ with the formula $g(n)= \zeta^{5n}$ for $ n \in \mathbb{Z}$. Show that $g$ is surjective.

I know that to prove a function is surjective I have to let $y\in \mu_{102}$ and find $x\in \mathbb{Z}$ such that $f(x) = y$. However, I'm not sure how to apply this to the given function.

Answer 1

If you take $n=45k$, $g(41k)=\zeta^{205k}=\zeta^{(204+1)k}=\zeta^k $.

Answer 2

Hints: 1) $\zeta$ generates $\mu_{102}$, so all you have to do is to find $x\in\mathbb{Z}$ such that $f(x) = \zeta$.

2) That means you want $\zeta^{5n} = \zeta$. If you write down what that means in terms of the exponents of $e$, you should be able to get a congruence that you can solve.