Determine all the generators of $\mathbb{Z}_{25}^{\times}$

Question

Determine all the generators of $\mathbb{Z}_{25}^{\times}$.

Is there some way that I can use the fact that $\mathbb{Z}_{25}^{\times}$ is cyclic generated by $3$?

Answer 1

Hint: Your intuition is correct - you can use the fact that $3$ is a generator.

An element $a \in \mathbb Z_{25}^\times$ is a generator if and only if $a$ has order $20= \left\vert\mathbb Z_{25}^\times\right\vert$ (i.e. $a^{20} = 1$ and $a^n \ne 1$ if $n <20$).

Also if $a \in \mathbb Z_{25}^\times$, then $a = 3^k$ for some $k$. Can you combine these facts to get a condition on $k$?