Finding the Greatest Common Factor (GCF)

Date: 03/31/98 at 21:14:18
From: Kimmy
Subject: I need help finding the GCF including exponents

Hi!

I am having trouble finding the greatest common factor (GCF) with 
exponents in the problems along with variables. Here is one of many 
questions that I am having trouble with:

     2     2
   3x y, 6x

Please help, 
Kimmy

Date: 04/21/98 at 17:39:15
From: Doctor Naomi
Subject: Re: I need help finding the GCF including exponents

Hi Kimmy!

Keep in mind that the GCF is the greatest common factor; that is, it 
is the biggest thing that is a factor of both expressions. When you 
write your expressions using exponents it is very easy (once you 
understand the idea) to find the GCF. 

Let's look at some examples. Let x^y mean "x to the y-power" (so 2^3 
means "2 to the 3rd power", or, "2 cubed" and equals 8).                 

Find the GCF of (a^2)(b^3)c and (a^2)(b^2)(c^5).

Because they are standing in for some unknown quantities, we do not 
know whether or not the variables a, b and c have common factors.  
There is no way for us to know what numbers these variables stand for, 
so we will treat each letter (variable) as a factor that can't be 
broken down further.

First let's look at the a's: a^2 is a factor of both expressions, so 
we'll include that in our GCF. 

Now let's look at the b's: b^3 is a factor of the first expression and 
b^2 is a factor of the second expression. Since b^2 is a factor of 
both b^2 and b^3, but b^3 is not a factor of b^2, the greatest common 
factor is b^2. 

Finally, we look at the c's: c is a factor of both expressions but no 
higher power of c will divide the first expression, so c is the GCF of 
c and c^5.  

Hence the GCF of (a^2)(b^3)c and (a^2)(b^2)(c^5) is (a^2)(b^2)c.  

Notice that we have "chosen" the lowest powers for each letter from 
the original expression to get our GCF. As long as you factor your 
original expressions completely, this method works well. It even works 
with finding the GCF of two numbers. 

For example: Find the GCF of 24 and 36.

You may be able to do this in your head, but with bigger numbers you 
may find that too difficult, so let's try to use the ideas from our 
a b c example. The first thing that you need to do is factor 24 and 18 
into primes. You have probably done factor trees or some similar 
method to do this. You should get 24 = 2*2*2*3 = (2^3) * 3 and 
36 = 2*2*3*3 = (2^2) * (3^2). So, we want to find the GCF of 
(2^3) * 3 and (2^2) * (3^2). First we look at the 2's: 2^2 is a factor 
of both expressions but no higher power of 2 is a factor of 2^2. Now 
for the 3's: 3 is a factor of both expressions but no higher power of 
3 is a factor of 3.  

So, the GCF of 24 and 36 is (2^2) * 3 = 12.

Now let's look at your question: find the GCF of 3(x^2)y and 6(x^2).

First you need to rewrite 6 as 2*3 so that we are looking at 3(x^2)y 
and 3*2(x^2). 3 is a factor of both expressions, 2 is not a factor of 
the first expression, so we can't include any 2's in our GCF, (x^2) is 
a factor of both expressions but no higher power of x is a factor of 
either expression and y is not a factor of the second expression so we 
can't include any y's in our GCF.  Thus, the GCF is 3(x^2).

Hope this helps!

-Doctors Naomi and Tracy,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/