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Let $R=\mathbb{Z}(i)$ be the ring of gaussian integers. Find the number of elements in the factor ring $R/A$ and describe the cosets if $A=R(1+3i)$.

I know the gaussian integers are $\mathbb{Z}(i)=\{a+bi\mid a,b\in\mathbb{Z}\}$.

So then a coset in $R/A$ would look like this: $\{ r+(1+3i)\mid r\in \mathbb{Z}(i)\}$ right? But aren't there infinitely many elements in $R/A$?

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$A$ also contains $(1+3i)(1-3i)=10$, so after reduction by $1+3i$ you can arrange to have only have coefficients of $i$ that are between $0$ and $2$, and constant terms between $0$ and $9$. That leaves only finitely many possibilities, which I'll let you count.

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