Proving you can write the Gaussian integers in a certain form Question: Show that any element of $\Bbb{Z}[i]$ can be written in the form $c+x$ where $c\in \{0,1,2,3,4\}$ and $x \in I$ where $I$ is the principal ideal $I= \langle 2-i\rangle$ generated by $2-i$
Answer:
I am given a hint to observe that $5=(2-i)(2+i) \in I$ but I don't see why I need it.
We have to prove that for any $a,b \in \Bbb Z$ 
$a+bi=c+x= c + (x+yi)\cdot(2-i)$
Which implies 
$a=c+2x+y$
$b=2y-x$
Doesn't this prove that I can write any member of $a+bi$ as a $c+x$?

Elementary solution: it is clear that every gaussian integer is $\equiv n\mod 2+i$ where $n\in \mathbb{Z}$. Further any integer is congruent to $0,1,2,3,4$  modulo $5$, and thus modulo $2+i$.
Conceptual solution: The number of residue classes modulo an algebraic number is $N\alpha$ where $N$ is the norm. Some take this as the definition, in which case it is theorem that this is the product of the conjugates. Thus in this case, residue field has order $N(2+i)=5$ and there is only one such field. 
