Let $R= \mathbb{Z}[i]$. In my class notes I have that $ R/(1+2i)R $ is either isomoprhic to $ \mathbb{Z}/5$ or $\mathbb{Z}/{5^2}$.
We know that $(1+2i)R$ is a prime ideal, so $R/(1+2i)R$ is an integral domains. The argument seemed to involve the kernel of the map $\mathbb{Z} \rightarrow R/(1+2i)R$.
The conclusion was that $ R/(1+2i)R \cong \mathbb{Z}/5.$
Similarly, $ R/3R $ was shown to have order 9. This proof seemed to rely on showing that $3R$ is a maximal ideal, which apparently automatically results in $ R/3R $ being a field of order 9.
I'm sorry this question is so vague, I don't understand what the argument was supposed to be.