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According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the terminology greatest common divisor.

However, the general definition used is that $d$ is said to be a GCD of $a$ and $b$ if

  1. $d$ divides both $a$ and $b$; and,
  2. If $d'$ also divides both $a$ and $b$, then $d'$ divides $d$.

My question is that why do we usually accept the second definition over the first. To me the first one seems very intuitive and simple, and does justice to the terminology. The same query goes for LCM as well.

Looking forward to your response. Thank you!

  • 9
    The general definition for GCD is applicable for those cases where you can't order things (how do you find a maximum, then?), like polynomials or Gaussian integers...2011-11-25
  • 11
    As J.M. points out, we cannot impose a linear order on several rings of interest, so there is no single "greatest element" in terms of "size" like in the case of integers. On the other hand, it seems natural to impose a *partial* order induced by divisibility; the GCD of $a$ and $b$ does correspond to the greatest of all common divisors of $a$ and $b$ according to this partial order.2011-11-25
  • 4
    It may do you good to prove (in the integers) that the two definitions are equivalent. Once you have done that, you can use either definition, whichever is more useful at the moment.2011-11-25
  • 1
    Reg. my previous comment: Well, it's strictly not a partial order if $a$ and $b$ can divide each other. Oh well... Let's ignore that for the purposes of my comment.2011-11-25
  • 0
    You _could_ impose a linear order on them, it's just not necessarily useful to do so.2011-11-25
  • 1
    With this definition, gcd and lcm are pullback and pushout, if you see $\mathbb{Z}$ as a category where $m\to n \iff m\mid n$ (every preorder is a category in this way). See http://en.wikipedia.org/wiki/Pullback_(category_theory). I don't think this is why this definition is preferred, but some (such as myself) find definitions that are categorical quite satisfying.2011-11-26

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