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- anonymous

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

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- anonymous

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

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- ganeshie8

I think that definition is the most intuitive one as it simply says what the name "least common multiple" means

- 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

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

and that's actually a theorem. I'm looking for a definition though

- ganeshie8

right, when you do prime factorization, that definition plays very nicely, you can play with the exponents

- 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

I don't see how that definition is any inferior to the definition of \(\gcd\)
what arithmetic structure do you have in mind ?

- anonymous

actually, that's exactly what i was looking for. Some sort of criteria that a least common multiple must follow

- ganeshie8

ohk.. but yes that definition is pretty useless when prime factorization is not possible

- anonymous

oh... I think it's still better than min{m : a|m and b|m}

- ganeshie8

both are same, aren't they..

- anonymous

they are indeed. I just prefer the one that is formally written out.

- ganeshie8

i see what you mean

- anonymous

thank you :D

- ganeshie8

np:)

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