If $G$ is a group with cardinality $pqr^2$ and $p,q,r$ different primes $\ge 2$, and given $1+np$ doesn't divide $qr^2$ for any $n \in \mathbb{Z}^+$,
How can you show $G$ is not simple?
I feel like you would use Sylow's Theorem but i'm not sure how to apply it to this scenario.
Let $n_p$ be the number of Sylow $p$-subgroups. By the Sylow theorems we have \begin{align*} n_p&\equiv 1\pmod p\\ n_p&\mid qr^2 \end{align*} By the first line we can write $n_p=1+np$ for some $n\in\Bbb N_0$. By assumption $1+np\nmid qr^2$ if $n>0$, hence $n=0$, i.e. $G$ has a unique Sylow $p$-subgroup and is therefore not simple.