How to show $2$ and $3$ are generators of the additive group $Z/(5)$.

Question

This may sound like an easy question, but I never learned cyclic groups due to my time constraint of my class and that the professor ran out of time to teach it. However, he left me with a problem to solve. The question is "Show that both $2$ and $3$ are generators of the additive group $Z/(5)$. I know that a cyclic group is a group that has one element that can generate other elements of the same form in that group. But how do I show with $2$ and $3$. I thought about showing $(2+3)^n$ would make that the generator of $Z/(5)$. Is that correct? If not, can someone explain to me how it works. Thanks.

Answer 1

Since

$$\Bbb Z=\langle 2,3\rangle\implies \Bbb Z/5\Bbb Z=\langle 2+5\Bbb Z,\,3+5\Bbb Z\rangle$$

Everything, of course, additive. Of course, you don't need two generators of either group, but you can use two generators.

Answer 2

Simplest proof: $2, 2+2=4, 2+2+2=1, 2+2+2+2=3, 2+2+2+2+2=0$. Hence $2$ generates the additive group $Z/(5)$, which has 5 elements. The proof that $3$ generates it is similar.