1. ## Prime Factorization

Here is what I need to prove:

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

I know that the necessary properties of [a,b] are
let a,b be positive integers. The LCM is a positive integer m such that,
(1) a|m & b|m
(2) for any n with a|n & b|n, we must have m|n

Any suggestions on proving this.

2. If you can prove that $(a,b)=1 \Rightarrow [a,b]=ab$ and $[da,db] = d[a,b]$ then

it will be short to get $(a,b)[a,b] = ab$

3. Let p be the prime factors of a and b.

$gcd(a,b)=p_1^{min \ a,b}*....p_k^{min \ a,b}$
$gcd(a,b)=p_1^{max \ a,b}*....p_k^{max \ a,b}$

What happens when you multiple the prime factorization together?