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 $c+x$?

${\rm mod}\,\ i\!-\!2\!:\,\  \color{#c00}{i\equiv 2}\,\Rightarrow\,0\equiv (\color{#c00}{2\!-\!i})(2\!+\!i)=\color{#0a0}{5},\ $ so $\,\ a +\color{#c00}i\,b\,\equiv\, (a+\color{#c00}2\,b) \bmod \color{#0a0}5\,\in\, \{0, 1, 2, 3, 4\}$
