Why do the integers modulo a composite positive number never form a simple group under addition?

Question

Let $p,q \in \mathbf{Z}$ such that $p > 1$ and $q > 1$. Then $\mathbf{Z}/pq\mathbf{Z}$ does not form a simple group under addition.

Why?

Answer 1

Because $\mathbb{Z}/(pq)\mathbb{Z}$ is abelian and has the proper, non-trivial cyclic subgroups $\langle p\rangle, \langle q\rangle$.

Answer 2

More generally,

$\mathbf{Z}/n\mathbf{Z}$ is a simple group iff $n$ is prime

Indeed, the subgroups of $\mathbf{Z}/n\mathbf{Z}$ are exactly he ones of the form $d\mathbf{Z}/n\mathbf{Z}$ where $d$ divides $n$.