Question: > 
> Show that, if $P$ and $Q$ are constants and $$y = P\cos(\ln(t)) + Q\sin(\ln(t))$$
>  then
>  $$t^2\frac{d^2y}{dt^2}+t\frac{dy}{dt}+y=0$$
> 
> 
>

Answer: 
Actually this is true only if $n\mid m$. Indeed
$$n\cdot\mathbf Z/m\mathbf Z=(n\mathbf Z+m\mathbf Z)/m\mathbf Z=\gcd(m,n)\mathbf Z/m\mathbf Z.$$
$(n)/(m)$ doesn't even make sense unless $(m) \subseteq (n)$, which is equivalent to $n | m$, so I'll assume this. But your statement is false unless $n = 0$ or $m = 0$. If neither is 0, then $(n)/(m) \cong \mathbb Z/(m/n)\mathbb Z$. This is a finite group, as $m/n \neq 0$. However, $(n)$ is infinite as $n \neq 0$ so these are not isomorphic.