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

Question

ganeshie8 · Answer

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

ganeshie8 · Answer

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 · Answer

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

anonymous · Answer

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

ganeshie8 · Answer

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

ganeshie8 · Answer

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 · Answer

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

anonymous · Answer

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

ganeshie8 · Answer

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

anonymous · Answer

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

ganeshie8 · Answer

both are same, aren't they..

anonymous · Answer

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

ganeshie8 · Answer

i see what you mean

anonymous · Answer

thank you :D

ganeshie8 · Answer

np:)