Thread: Prove: GCD(GCF) of (a,b) * LCM (a,b) = product a*b

Hello All,

I have begun working on this topic's proof but I am not sure where to go after this:

Given:

(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).

Prove:

(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,

Panglot

Originally Posted by panglot

Hello All,

I have begun working on this topic's proof but I am not sure where to go after this:

Given:

(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).

Prove:

(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,

Panglot

Here:

Product of GCD and LCM - ProofWiki