Homomorphic image of $\mathbb{Z}_{24}$ in $\mathbb{Z}_{18}$

Question

$\mathbb{Z}_m$ is homomorphic image of $\mathbb{Z}_{24}$ in $\mathbb{Z}_{18}$. Then which of the following cannot be the value of $m$?

1. 1
2. 2
3. 3
4. 4

My attempt: Answer is $4$. Because $4$ does not divide $18$. Am I correct?

Answer 1

Let $\varphi:(\Bbb Z_{24},+)\to \Bbb (Z_{18},+)$ a homomorphism of groups.

We know that

• The image of $\Bbb Z_{24}, \varphi(\Bbb Z_{24})$ is a sub-group of $\Bbb Z_{18}$;
• Let $n\in \Bbb N, n>1; n$ is the order of $\Bbb Z_n=\{\bar0,...,\overline{n-1}\}$;
• the order of a sub-group of $\Bbb Z_{18}$ divides $18$, so it belongs to $$\{1,2,3,6,9,18\}.$$

$4\notin \{1,2,3,6,9,18\}$. Yes you are correct, as Sir @AnneBauval told you :).