All element of the quotient ring $\mathbb{Z}_m/I$ for some $m \in \mathbb{N}$.

Question

Let $I = \lbrace \overline{0}, \overline{8}, \overline{16} \rbrace$ be an ideal in $\mathbb{Z}_{24}$. Find all elements of quotient ring $\mathbb{Z}_{24}/I$.

The answer is $\mathbb{Z}_{24}/I = \lbrace I, \overline{1} + I, \overline{2}+I, \dots, \overline{7} + I \rbrace$. But, I still can't understand how to obtain it. Anyone can explain, please? Thanks in advance.

Answer 1

It results directly from the Third isomorphism theorem: $I$ is simply the quotient $\:8\mathbf Z/24\mathbf Z$, so $$(\mathbf Z/24\mathbf Z)\big/I=(\mathbf Z/24\mathbf Z)\big/(8\mathbf Z/24\mathbf Z)\simeq \mathbf Z/8\mathbf Z.$$