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I have a book that says that 6 and -6 are both greatest common divisors of 12 and 18, and thus a gcd is not uniquely defined. I have an obvious question about this. 6 >-6 so how is -6 also a gcd?

I believe they say this because the definition of GCD given in the book includes a condition that if T is GCD of a and b, then any other number, say r, that divides a and b will also be a factor of T. Then, when including all integers, 6 and -6 both can be divided by all other common factors of 12 and 18, namely 2,3,1,-1,-2 and -3. So I'm wondering is the usage of the word "greatest" in "greatest common divisor" something different than that usual meaning of the word that refers to ordering of numbers, if we are taking this over all integers and not just natural numbers? Maybe in a precise definition a modulus will be involved?

Aniruddh
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    $-6$ is also a $\gcd$ because of its alternative definition, as you said. Every $d$ dividing $a$ and $b$ is a $\gcd$, if we have $d\mid c$ for every $c$ with $c\mid a$ and $c\mid b$. Usually we only consider this up to units in the ring, so then it is $+6$ and really the "greatest" common divisor of $12$ and $18$. This is for the ring $\Bbb Z$. And as Ethan said, for $R=\Bbb Z[i]$ a relation $6i<6$ doesn't make any sense. So the word "greatest" cannot be taken literally. – Dietrich Burde Apr 21 '22 at 19:15
  • Yes. I see the word "greatest" is not used in its usual English language sense. Which tells me the the condition specified is the functional part of the concept, not the word. I suppose I got confused because this is the first time I'm coming across this phenomena. – Aniruddh Apr 21 '22 at 19:23
  • The word "greatest" was meant for the integers, at least up to sign. For more general rings this is different. Something similar happens with prime numbers in $\Bbb Z$ and prime elements in a ring. A prime number would be called "irreducible" instead in more general rings. – Dietrich Burde Apr 21 '22 at 19:37
  • See esp. this answer in the linked dupe. – Bill Dubuque Apr 21 '22 at 19:44

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Good question. You have hit upon what will turn out to be the proper generalization in other rings.

The "greatest" that is the most useful is the one in the partial order on the integers determined by divisibility. Then among the common divisors of $a$ and $b$ there will be several that are greater than or equal to all the common divisors. Here "several" is just two, and one is the negative of the other. The positive one is the greatest in the usual order.

In the Gaussian integers (complex numbers of the form $a+bi$ for integers $a$ and $b$) any two elements will have four greatest common divisors of the form: $\pm g$ and $\pm ig$.

Ethan Bolker
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  • While the specific concepts you mentioned are quite above my station at this time, I have gained a sense of how mathematical thought is going to work as I continue my journey in mathematics. – Aniruddh Apr 21 '22 at 19:28
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    Those concepts - "ring" and "partial order" and "Gaussian integers" - are only a little above your station (not "quite above"). Enjoy the journey. – Ethan Bolker Apr 21 '22 at 19:33
  • Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Apr 21 '22 at 19:43
  • @BillDubuque I will try. Didn't think to search for a duplicate and thought it a thoughtful question. – Ethan Bolker Apr 21 '22 at 22:31