Finding Least Common Multiple (LCM) by Factoring

Date: 10/19/2001 at 01:13:45
From: Laura
Subject: Prime factoring

My math book has the example:

   Find the least common multiple of 6, 8, and 9.

     6 = 2 x 3   8 = 2 cubed    9 = 3 squared

     the least common multiple of 6, 8, and 9 is 
     2 cubed x 3 squared, or 72.

I understand that 8 x 9 = 72, but what happened to the 6 or 2 x 3?
I just don't get how to "figure prime factors and find the product of 
the greatest power of each prime factor."

Please help.
Laura

Date: 10/19/2001 at 19:08:35
From: Doctor Terrel
Subject: Re: Prime factoring

Hi Laura,

Yes, LCM is often a tricky thing to understand, especially going at it 
via prime factorizations first. Remember what LCM stands for in the 
beginning...

M = multiple     C = common      L = least (or lowest)

MULTIPLES:
   9 ==> 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, ...
   8 ==> 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, ...
   6 ==> 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, ...

COMMON:
Try to find a number or number(s) that appear in ALL 3 LINES. In the 
above sample, we only find 72. There are other larger ones, but it 
takes too long to write them. To do that in a fun way, go to The 
M.O.M. Game:

   http://www.geocities.com/ttrotter3/mom.html   

LEAST [smallest, first, etc.]
That has to be 72, this time.

Now via PRIME FACTORIZATION:

   For 6 it is 2 x 3;
   for 8 it is 2 x 2 x 2; and
   for 9 it is 3 x 3.

The task now before you is to make a collection or set - as small as 
possible - of those primes in which you can find all the factors of 
any given number.

LCM (6, 8, 9) = {2, 3, ...}         [6 is there so far]
              = {2, 3, 2, 2, ... }  [two more 2's were needed for 8]
              = {2, 3, 2, 2, 3}     [another 3 was needed for the 9]

   2 x 3 x 2 x 2 x 3 = 72.

   So LCM(6, 8, 9) = 72.

I hope this helps.

- Doctor Terrel, The Math Forum
  http://mathforum.org/dr.math/