Q:

Show that each of the $35$-roots of unity is the cube of some root of $35$th degree of unity.

After showing that the $n$-roots of unity form a multiplicative group I'm asked the following:

Show that each of the $35$-roots of unity is the cube of some root of $35$th
  degree of unity.

I don't have idea how to start with this problem. I'm only able to sort of formalize what the problem is asking me to prove, that is: 

Let $(\mathbb{C}^{35}, \cdot)$ be the group of the $35$-roots of
  unity. Then I have to show that $\forall a \in \mathbb{C}^{35}\
\exists\ b \in \mathbb{C}^{35}: b = a^3$.

Any help would be appreciated.

A:

$b=e^{\frac{2n\pi i}{35}}$ for some $n\in\{1,2,\dots,35\}$.
Then you are looking for $a=e^{\frac{2m\pi i}{35}}$, with $3m\equiv n, \mod35$.
There is always a solution to this since $3$ and $35$ are coprime.

