Elements of a factor group

Question

I believe the the elements of $3 \mathbb{Z}/12 \mathbb{Z}$ are $ \lbrace 0 +12 \mathbb{Z}, 3 +12 \mathbb{Z}, 6 +12 \mathbb{Z}, 9+12 \mathbb{Z} \rbrace = \langle 3 \rangle$, correct? But, what is the factor group isomorphic to and how would I know?

Answer 1

Note that $O(3+12\Bbb{Z})=4=O(G)$, where $G=\langle3 \rangle/\langle12 \rangle$ , so $\langle3 \rangle/\langle12 \rangle$ is cyclic and every cyclic group of order $4$ is isomorphic to $\Bbb{Z}_4$

In general, If $k$ divides $n$, then $\langle k \rangle/\langle n \rangle$ is a cyclic group of order $n/k$. So it is isomorphic to $Z_{n/k}$.