For simplicity, let us work only with positive integers.

We want to prove that if a,b,m are integers, then


First notice that any common multipleMathworldPlanetmath of ma and mb is also a multiple of m, so any common multiple of ma and mb is of the form mk with some integer k.

Now notice that if t=lcm(a,b) and u<t, it cannot happen that au and bu, since t is the smallest number, So, when au then mamu, and if bu then mbmu. We conclude that mu is not a common multiple of ma and mb when u<t.

So far, we proved that mt=mlcm(a,b) is a common multiple of ma and mb, and previous paragraph shows that there is no smaller common multiple, therefore mlcm(a,b) is the least common multiple of ma and mb, in other words:

Title lcm(ma,mb)=mlcm(a,b)
Canonical name mathrmlcmmambmmathrmlcmab
Date of creation 2013-03-22 15:03:22
Last modified on 2013-03-22 15:03:22
Owner drini (3)
Last modified by drini (3)
Numerical id 12
Author drini (3)
Entry type Theorem
Classification msc 11-00