Let a and b be positive integers, and let m be an integer such that Without using the prime factorization theorem, prove that by verifying that m satisfies the necessary properties of

Note that denotes the GCD of a and b and denotes the LCM of a and b.

Here is the definition for LCM that I have (the two properties that I need to show.)

A positive integer m is called the least common multiple of the nonzero integers a and b if

(i) m is a multiple of both a and b, and

(ii) any multiple of both a and b is also a multiple of m.

Property (i) was not very difficult to show, but I am struggling with the second one. Here is what I have so far.

Let and , where

Then and where

and

and

and

and

and

That is as much as I can get. How can I show that ?

Any hints would be appreciated.