Simple groups

Definition. A group is simple if its only normal subgroups are $\{e\}$ and $G$.

$\mathbb{Z}/p\mathbb{Z}$. Its only subgroups are $\{0\}$ and the whole group (since $p$ is prime). So it has no proper non-trivial normal subgroups. Simple.

$\mathbb{Z}/n\mathbb{Z}$ for composite $n$. If $d$ divides $n$ with $1<d<n$, the subgroup generated by $\overline{n/d}$ has order $d$. Since the group is abelian, this subgroup is normal. So the group is not simple.

Conclusion. $\mathbb{Z}/p\mathbb{Z}$ is simple for $p$ prime; $\mathbb{Z}/n\mathbb{Z}$ is not simple for $n$ composite.