TITLE: Find $n$ such that $J=n\mathbb Z$
QUESTION [6 upvotes]: Let the ideal $I=(4+3i)\mathbb Z[i]$ and $J=I\cap\mathbb Z$. Prove that $J$ is an ideal in $\mathbb Z,$ and find $n$ such that $J=n\mathbb Z$.

The first part of the task is to use the definition of an ideal by checking if it meets the appropriate properties. However I don't know what the ideal $J$ looks like because I must find part of the common $\mathbb Z$ and $\mathbb Z[i]$. 
Also, I don't know exactly how to find $n$?

REPLY [2 votes]: $$\overbrace{\dfrac{n}{4+3i} = \dfrac{(4-3i)n}{25}}^{\large {\rm rationalize\ denominator }_{\phantom |}\!\!}\in\Bbb Z[i]\iff 25\mid 4n,3n\iff 25\mid n = 4n\!-\!3n\qquad$$