Let a and b be positive integers, and let m be an integer such that $\displaystyle ab=m(a,b).$ Without using the prime factorization theorem, prove that $\displaystyle (a,b)[a,b]=ab$ by verifying that m satisfies the necessary properties of $\displaystyle [a,b].$

Note that $\displaystyle (a,b)$ denotes the GCD of a and b and $\displaystyle [a,b]$ 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 $\displaystyle a|q$ and $\displaystyle b|q$, where $\displaystyle q\in\mathbb{Z}.$

Then $\displaystyle q=ax$ and $\displaystyle q=by,$ where $\displaystyle x,y\in\mathbb{Z}$

$\displaystyle ==>\quad qb=abx\quad $ and $\displaystyle qa=aby$

$\displaystyle ==>\quad qb=mdx\quad $ and $\displaystyle qa=mdy$

$\displaystyle ==>\quad qhd=mdx\quad $ and $\displaystyle qkd=mdy$

$\displaystyle ==>\quad qh=mx\quad $ and $\displaystyle qk=my$

$\displaystyle ==>\quad m|qh\quad $ and $\displaystyle m|qk.$

That is as much as I can get. How can I show that $\displaystyle m|q$?

Any hints would be appreciated.