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anonymous

  • one year ago

what is an alternative definition for least common multiple besides min{m : a|m and b|m} ?

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  1. ganeshie8
    • one year ago
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    I think that definition is the most intuitive one as it simply says what the name "least common multiple" means

  2. ganeshie8
    • one year ago
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    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

  3. anonymous
    • one year ago
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    yeah, but It has no arithmetic structure. I was looking for something like i) m > 0 ii) a|m and b|m ...

  4. anonymous
    • one year ago
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    and that's actually a theorem. I'm looking for a definition though

  5. ganeshie8
    • one year ago
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    right, when you do prime factorization, that definition plays very nicely, you can play with the exponents

  6. ganeshie8
    • one year ago
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    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 \(b\mid c\) \(\implies\) \(m\le c\)

  7. ganeshie8
    • one year ago
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    I don't see how that definition is any inferior to the definition of \(\gcd\) what arithmetic structure do you have in mind ?

  8. anonymous
    • one year ago
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    actually, that's exactly what i was looking for. Some sort of criteria that a least common multiple must follow

  9. ganeshie8
    • one year ago
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    ohk.. but yes that definition is pretty useless when prime factorization is not possible

  10. anonymous
    • one year ago
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    oh... I think it's still better than min{m : a|m and b|m}

  11. ganeshie8
    • one year ago
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    both are same, aren't they..

  12. anonymous
    • one year ago
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    they are indeed. I just prefer the one that is formally written out.

  13. ganeshie8
    • one year ago
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    i see what you mean

  14. anonymous
    • one year ago
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    thank you :D

  15. ganeshie8
    • one year ago
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    np:)

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