# Compute the G.C.D of two Gaussian Integers

Tags:
1. Oct 22, 2015

### DeldotB

1. The problem statement, all variables and given/known data
Hello all I apologize for the triviality of this:
Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this:

Find the G.C.D of $a= 14+2i$ and $b=21+26i$.

$a,b \in \mathbb{Z} [ i ]$ - Gaussian Integers

2. Relevant equations

None

3. The attempt at a solution

Well, is it true that any common divisor must also divide the G.C.D of the norm's of $a$and$b$?

If so then, $norm(14+2i)=200$
$norm(21+26i)=1117$

Well, since 1117 and 200 are co-prime, their greatest common divisor is one. Thus,

Thus the G.C.D of a,b is a unit (1,-1,i,-i) in the ring.

Thanks

Last edited: Oct 22, 2015
2. Oct 22, 2015

### Staff: Mentor

Everything that divides (a+bi) also divides (a+bi)(a-bi), sure.
In the integers. You'll have to show that this is true for Gaussian integer factors as well.