AAR4

Find a g.c.d. of $6+7i\,$ and $12-3i\,$ in $\mathbb{Z}[i]\,$ by the Euclidean algorithm.

$|6+7i| = 36+49 = 85, |12-3i| = 144+9 = 153\,$

So divide $12-3i\,$ by $6+7i\,$ :

$12-3i = -i(6+7i)+(5+3i), |5+3i|<|6+7i|\,$

$6+7i=1(5+3i)+(1+4i), |1+4i|<|5+3i|\,$

$5+3i = 1(1+4i) + (4-i) = 1(1+4i)-i(1+4i) = (1-i)(1+4i)+0\,$

So the g.c.d. is $(1+4i)\,$ .