TITLE: How to calculate GCD of Gaussian integers?
QUESTION [30 upvotes]: Let $\mathbb Z [i] =\{a+bi: a,b \in \mathbb Z\}$.
What is the gcd of $11+7i$ and $18-i$ in $\mathbb Z [i]$?

REPLY [11 votes]: HINT $\ $ Modulo the gcd: $\ 11\: =\: {-}7\ i\ $ and $\ 18\: =\: i\:,\ $ therefore
$$ 11\ =\ (11-18)\ i\ =\ (-7\ i\:-\:i)\ i\ =\ 8\ \ \Rightarrow\ \ 3\: =\: 0\ \ \Rightarrow\ \ i\: =\: 6\cdot 3\: =\: 0\ \ \Rightarrow\ \ 1\: =\: 0$$