Math Is Fun Forum

mortage762

Member

Registered: 2007-05-04

Posts: 6

LCM and GCD Question

I had to figure out the following lcms and gcds and compare the two:

I got that
LCM(8,12)=24 and GCD(8,12)=4
LCM(20,30)=60 and GCD(20,30)=10
LCM(51,68)=204 and GCD(51,68)=17
LCM(23,18)=414 and GCD(23,18)=1

The pattern I found was that if you take the first number enclosed in the parentheses, divide it by the GCD, and multiply it by the second number in the parentheses, you get the LCM. For example, 8/4=2, which when multiplied by 12, gives you the LCM of 24.

My question is, how do I show that this relationship holds true for all m and n (where m and n are the numbers between the parentheses)?

Thanks in advance, everyone!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: LCM and GCD Question

So you got:

Let's take this and rewrite it a bit:

Now do you understand the fundamental theorem of arithmetic?  That every integer can be expressed as a unique product of primes?  If you do, apply this to figure out what the gcd and lcm would be.  If not, I'm not sure how to approach this problem.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: LCM and GCD Question

You can prove this simpler result first: If gcd(a,b) = 1, then lcm(a,b) = ab.

Proof:

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: LCM and GCD Question

I never did like that proof.  There is nothing wrong with it, technically, but it sidesteps the real picture.  GCD is all all the primes that n and m have in common multiplied together.  LCM eliminates all the ones they have in common.  So multiplying them together makes all the prime factors of n and m, which of course is n*m.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: LCM and GCD Question

I AM BUMPING THIS THREAD BECAUSE I FEEL SOME PEOPLE REALLY NEED TO LOOK AT IT.