For each number $d$ dividing 12, list the a's with $1 \leq a < 13$ and $e_{13} (a) = d$

Question

For each number $d$ dividing 12, list the a's with $1 \leq a < 13$ and $e_{13} (a) = d$

Can some explain the method of solving this number theory problem. Giving me a hard time, thanks.

Answer 1

Take any $a\neq 1$, for example, $2$. Now compute the successive powers of $a$: $$2,4,8,3,6,12,11,9,5,10,7,1$$ We have been lucky: $2$ is a primitive root, that is, the powers of $2$ span every element of $\Bbb Z_{13}$.

Now, what would be $e(4)=e(2^2)$? And $e(5)=e(2^9)?$