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Compute the G.C.D of two Gaussian Integers

  1. Oct 22, 2015 #1
    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 [itex]a= 14+2i [/itex] and [itex]b=21+26i [/itex].

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

    2. Relevant equations


    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 [itex] a [/itex]and[itex] b [/itex]?

    If so then, [itex] norm(14+2i)=200 [/itex]
    [itex]norm(21+26i)=1117 [/itex]

    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.

    Last edited: Oct 22, 2015
  2. jcsd
  3. Oct 22, 2015 #2


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    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.
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