AAR4

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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)\,.

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