anonymous
  • anonymous
what is an alternative definition for least common multiple besides min{m : a|m and b|m} ?
Mathematics
schrodinger
  • schrodinger
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ganeshie8
  • ganeshie8
I think that definition is the most intuitive one as it simply says what the name "least common multiple" means
ganeshie8
  • ganeshie8
for two integers \(a,b\) we do have this relationship though : \[\text{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}\] but that doesn't work for more than two integers
anonymous
  • anonymous
yeah, but It has no arithmetic structure. I was looking for something like i) m > 0 ii) a|m and b|m ...

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anonymous
  • anonymous
and that's actually a theorem. I'm looking for a definition though
ganeshie8
  • ganeshie8
right, when you do prime factorization, that definition plays very nicely, you can play with the exponents
ganeshie8
  • ganeshie8
that same definition can be expressed as below : \(\text{lcm}(a,b)\) is the positive integer \(m\) satisfying : 1) \(a\mid m\) and \(b\mid m\) 2) \(a\mid c\) and \(b\mid c\) \(\implies\) \(m\le c\)
ganeshie8
  • ganeshie8
I don't see how that definition is any inferior to the definition of \(\gcd\) what arithmetic structure do you have in mind ?
anonymous
  • anonymous
actually, that's exactly what i was looking for. Some sort of criteria that a least common multiple must follow
ganeshie8
  • ganeshie8
ohk.. but yes that definition is pretty useless when prime factorization is not possible
anonymous
  • anonymous
oh... I think it's still better than min{m : a|m and b|m}
ganeshie8
  • ganeshie8
both are same, aren't they..
anonymous
  • anonymous
they are indeed. I just prefer the one that is formally written out.
ganeshie8
  • ganeshie8
i see what you mean
anonymous
  • anonymous
thank you :D
ganeshie8
  • ganeshie8
np:)

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