I have begun working on this topic's proof but I am not sure where to go after this:
(a*b), a,b elements of the natural numbers
There exists d such that d is the GCD (a,b) and is a divisor of a and a divisor of b.
There exists e such that e is the LCM (a,b).
(e*d) = (a*b)
At this point, I want to use the Well-Ordering Principle to prove that the set of multiples contains a smallest possible multiple, but I am not sure how to prove this through contradiction. Would there be a way to do this, and then to prove the initial premise?
Thank you very much for your time,
Product of GCD and LCM - ProofWiki