Least Common Multiple

Date: 3/29/96 at 11:8:1
From: JONATHAN FIELDS
Subject: least common multiples

Hi Dr. Math.  

The question I have is about how to find the least common
multiples from a set of numbers.

I know that a multiple of a number has that number as one of its 
factors.

What is the least common multiple for the numbers 5 and 25?  

Is there a stated rule or definition for least common multiples?
Thanks! 

Date: 3/30/96 at 16:57:26
From: Doctor Sebastien
Subject: Re: least common multiples

Hi,

The least common multiple (L.C.M.) of a set of numbers is the smallest 
number that is a multiple of all numbers in the set.

The way I was told to find the LCM of a set of numbers was

  1. align them in a line.
  2. Then divide all numbers by an appropriate integer.  
  3. If any number is not divisible by that integer, don't divide that 
      number and leave it untouched.
  4. Repeat steps 2 and 3 until all 1's are obtained.
  5. Then multiply all the divisors together.  The answer is the LCM.

To better explain what I mean, I will find the LCM of 5 and 25.

STEP 1: 5, 25

            1,  5
          -------
STEP 2: 5 | 5, 25

            1,  1
          --------
STEP 2: 5 | 1,  5

In the previous step, the 1 is left untouched because it is not 
divisible by 5.  The LCM is then 5*5 = 25.

I hope I have been helpful.

-Doctor Sebastien,  The Math Forum

Date: 3/30/96 at 17:7:18
From: Doctor Ken
Subject: Re: Dr. Math -- least common multiples

Hello!

Here's another way to think about doing Least Common Multiples:

Since the least common multiple of two numbers is the smallest 
number that is a multiple of both the other numbers, you could 
also think of that as being the smallest number that contains all the 
_factors_ of the first two numbers.  So let's say we want to find 
the LCM of 28 and 42:

Well, let's list all the prime factors: 
      28 = 2*2*7
      42 = 2*3*7

So our LCM is going to have to be divisible by 4 (which is 2*2), 3, 
and 7. Do you see why?  So the LCM of 28 and 42 is 4*3*7 = 84.

Note that if we write our numbers as
      28 = 2*2*7  =  2^2 * 3^0  * 7^1
      42 = 2*3*7  =  2^1 * 3^1  * 7^1,

Then we can just take the biggest exponents and call that our number:
              LCM =  2^2 * 3^1 * 7^1  =  84.

Hope that helps!

-Doctor Ken,  The Math Forum