Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$ The question is : 

Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic.

My try:

I started with elements $2, 1+i \in \mathbb Z[i]$. Clearly, the product is $0$ in the coset representation.

The part where i am struck at is the number of elements and the characteristic.

Well, $2+2i+\langle 2+2i\rangle=0+\langle 2+2i\rangle$

Can we directly write $i^2=-1$ in the coset representation?
The following is not the most elementary approach, but it puts the question in a nice perspective (in my humble opinion):

There is an isomorphism $\Bbb{Z}[i]\cong\Bbb{Z}[X]/(X^2+1)$, given by $i\ \mapsto\ X$, yielding an isomorphism
$$\Bbb{Z}[i]/(2+2i)\cong\Bbb{Z}[X]/(X^2+1,2+2X).$$
The characteristic of the latter the least positive integer in the ideal $(X^2+1,2+2X)\cap\Bbb{Z}$ of $\Bbb{Z}$. Its number of elements is precisely the index of the ideal $(X^2+1,2+2X)$ in $\Bbb{Z}[X]$, which is the same as the resultant of $X^2+1$ and $2+2X$, which is in itself a nice fact to prove.
I usually solve it by drawing the square generated by $2+2i$.  

....|....
..xbbbx..
..aoooa..
--aoooa--
..aoooa..
..xbbbx..
....|....


I count 16 points:


the 9 points marked o are distinct
the 3 points marked a on the left and the right are congruent mod $(2+i)$
the 3 points marked b on the top and bottom are congruent mod $(2+i)
the 1 points marked x are the same