6.1: The Greatest Common Factor
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We begin this section with definitions of factors and divisors. Because 24=2⋅12, both 2 and 12 are factors of 24. However, note that 2 is also a divisor of 24, because when you divide 24 by 2 you get 12, with a remainder of zero. Similarly, 12 is also a divisor of 24, because when you divide 24 by 12 you get 2, with a remainder of zero.
Definitions: Factors and Divisors
Suppose m and n are integers. Then m is a divisor (factor) of n if and only if there exists another integer k so that n=m⋅k.
The words divisor and factor are equivalent. They have the same meaning.
Example 6.1.1
List the positive divisors (factors) of 24.
Solution
First, list all possible ways that we can express 24 as a product of two positive integers:
24=1⋅24 or 24=2⋅12 or 24=3⋅8 or 24=4⋅6
Therefore, the positive divisors (factors) of 24 are 1,2,3,4,6,8, and 24.
Exercise 6.1.1
List the positive divisors of 18.
- Answer
-
1,2,3,6,9, and 18
Example 6.1.2
List the positive divisors (factors) that 36 and 48 have in common.
Solution
First, list all positive divisors (factors) of 36 and 48 separately, then box the divisors that are in common.
Divisors of 36 are: [1],[2],[3],[4],[6],9,[12],18,36
Divisors of 48 are: [1],[2],[3],[4],[6],8,[12],16,24,48
Therefore, the common positive divisors (factors) of 36 and 48 are 1,2,3,4,6, and 12.
Exercise 6.1.2
List the positive divisors that 40 and 60 have in common.
- Answer
-
1,2,4,5,10, and 20
Definition: Greatest Common Divisor
The greatest common divisor (factor) of a and b is the largest positive number that divides evenly (no remainder) both a and b. The greatest common divisor of a and b is denoted by the symbolism GCD(a,b). We will also use the abbreviation GCF(a,b) to represents the greatest common factor of a and b.
Remember, greatest common divisor and greatest common factor have the same meaning. In Example 6.1.2, we listed the common positive divisors of 36 and 48. The largest of these common divisors was 12. Hence, the greatest common divisor (factor) of 36 and 48 is 12, written GCD(36,48)=12.
With smaller numbers, it is usually easy to identify the greatest common divisor (factor).
Example 6.1.3
State the greatest common divisor (factor) of each of the following pairs of numbers:
- 18 and 24
- 30 and 40
- 16 and 24
Solution
In each case, we must find the largest possible positive integer that divides evenly into both the given numbers.
- The largest positive integer that divides evenly into both 18 and 24 is 6. Thus, GCD(18,24)=6.
- The largest positive integer that divides evenly into both 30 and 40 is 10. Thus, GCD(30,40)=10.
- The largest positive integer that divides evenly into both 16 and 24 is 8. Thus, GCD(16,24)=8.
Exercise 6.1.3
State the greatest common divisor of 36 and 60.
- Answer
-
12
With larger numbers, it is harder to identify the greatest common divisor (factor). However, prime factorization will save the day!
Example 6.1.4
Find the greatest common divisor (factor) of 360 and 756.
Solution
Prime factor 360 and 756, writing your answer in exponential form.

Thus:
360=23⋅32⋅5756=22⋅33⋅7
Note
To find the greatest common divisor (factor), list each factor that appears in common to the highest power that appears in common.
In this case, the factors 2 and 3 appear in common, with 22 being the highest power of 2 and 32 being the highest power of 3 that appear in common. Therefore, the greatest common divisor of 360 and 756 is:
GCD(360,756)=22⋅32=4⋅9=36
Therefore, the greatest common divisor (factor) is GCD(360,756)=36. Note what happens when we write each of the given numbers as a product of the greatest common factor and a second factor:
360=36⋅10756=36⋅21
In each case, note how the second second factors (10 and 21) contain no additional common factors.
Exercise 6.1.4
Find the greatest common divisor of 120 and 450.
- Answer
-
30
Finding the Greatest Common Factor of Monomials
Example 6.1.4 reveals the technique used to find the greatest common factor of two or more monomials.
Finding the GCF of two or more monomials
To find the greatest common factor of two or more monomials, proceed as follows:
- Find the greatest common factor (divisor) of the coefficients of the given monomials. Use prime factorization if necessary.
- List each variable that appears in common in the given monomials.
- Raise each variable that appears in common to the highest power that appears in common among the given monomials.
Example 6.1.5
Find the greatest common factor of 6x3y3 and 9x2y5.
Solution
To find the GCF of 6x3y3 and 9x2y5, we note that:
- The greatest common factor (divisor) of 6 and 9 is 3.
- The monomials 6x3y3 and 9x2y5 have the variables x and y in common.
- The highest power of x in common is x2. The highest power of y in common is y3.
Thus, the greatest common factor is GCF(6x3y3,9x2y5)=3x2y3. Note what happens when we write each of the given monomials as a product of the greatest common factor and a second monomial:
6x3y3=3x2y3⋅2x9x2y5=3x2y3⋅3y
Observe that the set of second monomial factors (2x and 3y) contain no additional common factors.
Exercise 6.1.5
Find the greatest common factor of 16xy3 and 12x4y2.
- Answer
-
4xy2
Example 6.1.6
Find the greatest common factor of 12x4, 18x3, and 30x2.
Solution
To find the GCF of 12x4, 18x3, and 30x2, we note that:
- The greatest common factor (divisor) of 12, 18, and 30 is 6.
- The monomials 12x4, 18x3, and 30x2 have the variable x in common.
- The highest power of x in common is x2.
Thus, the greatest common factor is GCF(12x4,18x3,30x2)=6x2. Note what happens when we write each of the given monomials as a product of the greatest common factor and a second monomial:
12x4=6x2⋅2x218x3=6x2⋅3x30x2=6x2⋅5
Observe that the set of second monomial factors (2x2, 3x, and 5) contain no additional common factors.
Exercise 6.1.6
Find the greatest common factor of 6y3, 15y2, and 9y5.
- Answer
-
3y2
Factor Out the GCF
In Chapter 5, we multiplied a monomial and polynomial by distributing the monomial times each term in the polynomial.
2x(3x2+4x−7)=2x⋅3x2+2x⋅4x−2x⋅7=6x3+8x2−14x
In this section we reverse that multiplication process. We present you with the final product and ask you to bring back the original multiplication problem. In the case 6x3+8x2−14x, the greatest common factor of 6x3, 8x2, and 14x is 2x. We then use the distributive property to factor out 2x from each term of the polynomial.
6x3+8x2−14x=2x⋅3x2+2x⋅4x−2x⋅7=2x(3x2+4x−7)
Factoring
Factoring is “unmultiplying.” You are given the product, then asked to find the original multiplication problem.
First rule of factoring
If the terms of the given polynomial have a greatest common factor (GCF), then factor out the GCF.
Let’s look at a few examples that factor out the GCF.
Example 6.1.7
Factor: 6x2+10x+14
Solution
The greatest common factor (GCF) of 6x2, 10x and 14 is 2. Factor out the GCF.
6x2+10x+14x=2⋅3x2+2⋅5x+2⋅7=2(3x2+5x+7)
Checking your work
Every time you factor a polynomial, remultiply to check your work.
Check: Multiply. Distribute the 2.
2(3x2+5x+7)=2⋅3x2+2⋅5x+2⋅7=6x2+10x+14
That’s the original polynomial, so we factored correctly.
Exercise 6.1.7
Factor: 9y2−15y+12
- Answer
-
3(3y2−5y+4)
Example 6.1.8
Factor: 12y5−32y4+8y2
Solution
The greatest common factor (GCF) of 12y5, 32y4 and 8y2 is 4y2. Factor out the GCF.
12y5−32y4+8y2=4y2⋅3y3−4y2⋅8y2+4y2⋅2=4y2(3y3−8y2+2)
Check: Multiply. Distribute the monomial 4y2.
4y2(3y3−8y2+2)=4y2⋅3y3−4y2⋅8y2+4y2⋅2=12y5−32y4+8y2
That’s the original polynomial. We have factored correctly.
Exercise 6.1.8
Factor: 8x6+20x4−24x3
- Answer
-
4x3(2x3+5x−6)
Example 6.1.9
Factor: 12a3b+24a2b2+12ab3
Solution
The greatest common factor (GCF) of 12a3b, 24a2b2 and 12ab3 is 12ab. Factor out the GCF.
12a3b+24a2b2+12ab3=12ab⋅a2−12ab⋅2ab+12ab⋅b2=12ab(a2+2ab+b2)
Check: Multiply. Distribute the monomial 12ab.
12ab(a2+2ab+b2)=12ab⋅a2−12ab⋅2ab+12ab⋅b2=12a3b+24a2b2+12ab3
That’s the original polynomial. We have factored correctly.
Exercise 6.1.9
Factor: 15s2t4+6s3t2+9s2t2
- Answer
-
3s2t2(5t2+2s+3)
Speeding Things Up a Bit
Eventually, after showing your work on a number of examples such as those in Examples 6.1.7, 6.1.8, and 6.1.9, you’ll need to learn how to perform the process mentally.
Example 6.1.10
Factor each of the following polynomials:
- 24x+32
- 5x3−10x2−10x
- 2x4y+2x3y2−6x2y3
Solution
In each case, factor out the greatest common factor (GCF):
- The GCF of 24x and 32 is 8. Thus,24x+32=8(3x+4)
- The GCF of 5x3, 10x2, and 10x is 5x. Thus: 5x3−10x2−10x=5x(x2−2x−2)
- The GCF of 2x4y, 2x3y2, and 6x2y3 is 2x2y. Thus:2x4y+2x3y2−6x2y3=2x2y(x2+xy−3y2)
As you speed things up by mentally factoring out the \(\mathrm{GCF}\), it is even more important that you check your results. The check can also be done mentally. For example, in checking the third result, mentally distribute 2x2y times each term of x2+xy−3y2. Multiplying 2x2y times the first term x2 produces 2x4y, the first term in the original polynomial.
Continue in this manner, mentally checking the product of 2x2y with each term of x2+xy−3y2, making sure that each result agrees with the corresponding term of the original polynomial.
Exercise 6.1.10
Factor: 18p5q4−30p4q5+42p3q6
- Answer
-
6p3q4(3p2−5pq+7q2)
Remember that the distributive property allows us to pull the GCF out in front of the expression or to pull it out in back. In symbols:
ab+ac=a(b+c) or ba+ca=(b+c)a
Example 6.1.11
Factor: 2x(3x+2)+5(3x+2)
Solution
In this case, the greatest common factor (GCF) is 3x+2.
2x(3x+2)+5(3x+2)=2x⋅(3x+2)+5⋅(3x+2)=(2x+5)(3x+2)
Because of the commutative property of multiplication, it is equally valid to pull the GCF out in front.
2x(3x+2)+5(3x+2)=2x⋅(3x+2)+5⋅(3x+2)=(3x+2)(2x+5)
Note that the order of factors differs from the first solution, but because of the commutative property of multiplication, the order does not matter. The answers are the same.
Exercise 6.1.11
Factor: 3x2(4x−7)+8(4x−7)
- Answer
-
(3x2+8)(4x−7)
Example 6.1.12
Factor: 15a(a+b)−12(a+b)
Solution
In this case, the greatest common factor (GCF) is 3(a+b).
15a(a+b)−12(a+b)=3(a+b)⋅5a−3(a+b)⋅4=3(a+b)(5a−4)
Alternate solution:
It is possible that you might fail to notice that 15 and 12 are divisible by 3, factoring out only the common factor a+b.
15a(a+b)−12(a+b)=15a⋅(a+b)−12⋅(a+b)=(15a−12)(a+b)
However, you now need to notice that you can continue, factoring out 3 from both 15a and 12.
=3(5a−4)(a+b)
Note that the order of factors differs from the first solution, but because of the commutative property of multiplication, the order does not matter. The answers are the same.
Exercise 6.1.12
Factor: 24m(m−2n)+20(m−2n)
- Answer
-
4(6m+5)(m−2n)
Factoring by Grouping
The final factoring skill in this section involves four-term expressions. The technique for factoring a four-term expression is called factoring by grouping.
Example 6.1.13
Factor by grouping: x2+8x+3x+24
Solution
We “group” the first and second terms, noting that we can factor an x out of both of these terms. Then we “group” the third and fourth terms, noting that we can factor 3 out of both of these terms.
Now we can factor x+8 out of both of these terms.
(x+3)(x+8)
Exercise 6.1.13
Factor by grouping: x2−6x+2x−12
- Answer
-
(x+2)(x−6)
Let’s try a grouping that contains some negative signs.
Example 6.1.14
Factor by grouping: x2+4x−7x−28
Solution
We “group” the first and second terms, noting that we can factor x out of both of these terms. Then we “group” the third and fourth terms, then try to factor a 7 out of both these terms.
This does not lead to a common factor. Let’s try again, this time factoring a −7 out of the third and fourth terms.
That worked! We now factor out a common factor x+4.
(x−7)(x+4)
Exercise 6.1.14
Factor by grouping: x2−5x−4x+20
- Answer
-
(x−4)(x−5)
Let’s increase the size of the numbers a bit.
Example 6.1.15
Factor by grouping: 6x2−8x+9x−12
Solution
Note that we can factor 2x out of the first two terms and 3 out of the second two terms.
Now we have a common factor 3x−4 which we can factor out.
(2x+3)(3x−4)
Exercise 6.1.15
Factor by grouping: 15x2+9x+10x+6
- Answer
-
(3x+2)(5x+3)
As the numbers get larger and larger, you need to factor out the (GCF) from each grouping. If not, you won’t get a common factor to finish the factoring.
Example 6.1.16
Factor by grouping: 24x2−32x−45x+60
Solution
Suppose that we factor 8x out of the first two terms and −5 out of the second two terms.
That did not work, as we don’t have a common factor to complete the factoring process. However, note that we can still factor out a 3 from 9x−12. As we’ve already factored out a 5, and now we see can factor out an additional 3, this means that we should have factored out 3 times 5, or 15, to begin with. Let’s start again, only this time we’ll factor 15 out of the second two terms.
Beautiful! We can now factor out 3x−4.
(8x−15)(3x−4)
Exercise 6.1.16
Factor by grouping: 36x2−84x+15x−35
- Answer
-
(12x+5)(3x−7)