Finding LCDs, LCMs, and GCFs

Date: 06/09/98 at 20:50:15
From: Cheryl
Subject: LCD, LCM, GCF

I don't know how to do the greatest common factor, the least common 
multiple, and the least common denominator.

Date: 06/09/98 at 21:21:57
From: Doctor Gary
Subject: Re: LCD, LCM, GCF

Let's focus on what you do know, and maybe we'll discover that you 
know more than you think you do.

Do you know how to express a number as the product of its prime 
factors? For example, do you know that 120 can be expressed (using x 
to indicate multiplication) as 2 x 2 x 2 x 3 x 5?   

There are some very useful shortcuts for factoring a number. You 
should be familiar with how to tell if a number is divisible by 2 or 5 
just by looking at the last digit. (If the last digit is 0, 2, 4, 6, 
or 8, it's divisible by 2. If the last digit is 0 or 5, it's divisible 
by 5.) You should also know that numbers are divisible by 3 only when 
the sum of the digits of the number is divisible by 3.

When you do factor, it's always a very good idea to list the factors 
in increasing order, because it makes it much easier to see "common" 
factors.  

GCF
---
The greatest common factor of two (or more) numbers is the product of 
all the factors the numbers have in common. If you wanted to find the 
greatest common factor of 32 and 76, you would express both as 
products of their prime factors, and look for factors common to both:

  32  =  2 x 2 x 2 x 2 x 2

  76  =  2 x 2 x 19

There are two 2s common to both numbers, so 2 x 2 = 4 is the "greatest 
common factor" of 32 and 76.

LCM
---
The least common multiple of two (or more) numbers is the product of 
one number times the factors of the other number(s) that aren't 
common.

If you wanted to find the least common multiple of 32 and 76, you'd 
multiply 32 by 19, because 19 is the only factor of 76 that isn't 
common to the factors of 32.

LCD
---
Least common denominators is just a fancy way of saying least common 
multiple for two (or more) different denominators, so if you know how 
to find least common multiples, you can find least common 
denominators. Once you've found a least common denominator, you 
re-express each fraction by multiplying by a carefully chosen fraction 
which is equal to 1 (remember that multiplying by 1 doesn't change the 
value of a number).

For example, in order to add 1/6 and 1/8, we find the least common 
multiple of the denominators, which is 24. Then we multiply 1/6 by 4/4 
and multiply 1/8 by 3/3, to re-express each addend as some number of 
"24ths":

   1/6 x 4/4  =  4/24
   1/8 x 3/3  =  3/24

   1/6 + 1/8  =  4/24 + 3/24  =  (4+3)/24  =  7/24  

Remember, whether you're trying to find the greatest common factor (a 
fancy way of saying biggest number by which two or more numbers are 
divisible), or the least common multiple (a fancy way of saying the 
smallest number which is divisible by two or more numbers), the key is 
factoring the numbers to their prime factors, and then noting which 
factor(s) the numbers have in common. 

The product of the common factors will be the greatest common factor.  
The product of one number and the factors of the other that are NOT 
common to the first will be the least common multiple.

Enjoy. Once you understand it, it's actually sort of fun. 

-Doctor Gary,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/