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Homework Help: Mathematics: Algebra: Greatest Common Factor (GCF)
Finding the Greatest Common Factor (or GCF) of two or more numbers or algebraic
terms is an important process which is explained in this lesson. This process
will be used in later lessons on factoring a GCF from an algebraic expression,
and on simplifying fractions.
In the first example, we ill find the GCF of two numbers, 15 and 30.
Begin by writing down all of the positive factors of each number.
| 15 |
(1, 3, 5, 15) |
| 30 |
(1, 2, 3, 5, 6, 10, 15, 30) |
Next, mark all of the factors that both 15 and 30 have in common.
In the example below, we marked the numbers by using bold text and
highlighting. It is usually more convenient to circle or underline the numbers
when working problems out on paper.
| 15 |
(1, 3, 5, 15)
|
| 30 |
(1, 2, 3, 5, 6, 10, 15, 30)
|
The highest number, that both sets of factors have in common is the GCF. In this case, the GCF is 15.
Finding the GCF of two terms that contain variables, like the two terms below,
is covered on the next page.
14j2k3
21j
We will now present an example of finding the GCF of two algebraic terms:
14j2k3
21j
First, determine the numeric coefficient of each term:
14
21
Using the method shown earlier, find the GCF of each coefficient:
| 14 |
(1, 2, 7, 14)
|
| 21 |
(1, 3, 7, 21)
|
Greatest Common Factor / GCF = 7
Now find the smallest exponent of each variable. For the variable j we have
an exponent of 2 and an exponent of 1 (recall that a variable has an exponent
of 1 if the exponent isn't explicitly shown). Thus, the lowest exponent for j
is 1. So far we have
j1 or j
Now recall that in a term where a given variable is not present, the variable
has an exponent of 0. Thus for the variable k we have k3 and k0.
As a result, we now have
jk0 or simply j
Thus, the GCF of the variables from each term is j.
Now the GCF of the two terms is the GCF of the coefficients times
the GCF of the variables. So simply write the GCF of the coefficients (numbers)
with the GCF of the letters
7j
Often, you will need to find the GCF of three or more terms.
The methods are an extension of the methods presented for two terms.
Instead of finding the greatest numeric factor common among two terms,
you find the greatest factor that is common among the 3 or more terms. And
instead of finding the lowest exponent of a given variable between two terms,
find the lowest exponent of a given variable among the three or more terms.
The work for finding the GCF of three terms is shown below.
102k5m2
51k4m
153k2m2
First find the GCF of the coefficients:
| 102 |
(1, 2, 3, 6, 17, 34, 51, 102)
|
| 51 |
(1, 3, 17, 51)
|
| 153 |
(1, 3, 9, 17, 51, 153)
|
GCF (of coefficients only) = 51
Next find the GCF of the variables:
k5m2
k4m
k2m2
GCF (of variables) = k2m
Now multiply the two GCFs
GCF of the entire term = 51k2m
Homework Help: Mathematics: Algebra
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