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.