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

1. ganeshie8

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

2. 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

3. anonymous

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

4. anonymous

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

5. ganeshie8

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

6. 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 $$b\mid c$$ $$\implies$$ $$m\le c$$

7. ganeshie8

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

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

9. ganeshie8

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

10. anonymous

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

11. ganeshie8

both are same, aren't they..

12. anonymous

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

13. ganeshie8

i see what you mean

14. anonymous

thank you :D

15. ganeshie8

np:)