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Greatest Common Factor
Goal: Find the greatest common factor of two or more numbers.
Vocabulary
A whole number that is a factor of two or more
nonzero whole numbers.
Common factor:
The GCF is
sometimes called the
greatest common divisor
(GCD) because it is the
largest common factor
that can be divided
evenly into the
given numbers.
Greatest common factor (GCF):
Relatively prime :
EXAMPLE
1
The greatest of the common factors
of two or more numbers.
Two numbers are relatively prime if their
greatest common factor is 1.
Finding the Greatest Common Factor
Find the greatest common factor of 45 and 60.
Begin by writing the prime factorization of each number. Find the product
of the common prime factors.
45 = 3 3 5
60 = 2 2 3 5
The common prime factors are 3 and 5 . The GCF of 45 and 60 is
the product of these factors.
ANSWER The GCF of 45 and 60 is 3
EXAMPLE
2
p
5 , or
15 .
Identifying Relatively Prime Numbers
Decide whether the numbers 88 and 63 are relatively prime. If they
are not relatively prime, find the greatest common factor.
Begin by writing the prime factorization of each number. Then find the
product of the common prime factors.
88 23 p 11
63 32 p 7
There are no common prime factors. However, two numbers always
have 1 as a common factor. So, the GCF is 1.
ANSWER The numbers 88 and 63 are relatively prime .
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Y our turn now
1. 16, 20
Find the greatest common factor of the numbers.
2. 28, 56
4
28
3. 36, 63
4. 72, 126
9
18
Decide whether the numbers are relatively prime. If they are not relatively
prime, find the GCF.
5. 24, 64
6. 105, 115
No; GCF: 8
EXAMPLE
3
No; GCF: 5
7. 32, 45
8. 140, 81
Yes
Yes
Finding the GCF of Monomials
Find the greatest common factor of 15x 3 and 10x 4.
First factor each expression.
15x 3 3 p 5 p x p x p x
10x 4 2 p 5 p x p x p x p x
The common factors are 5 and x 3 . The GCF is the product of
the common factors.
ANSWER The GCF is 5x 3 .
Y our turn now
9. 3a, 21a
3a
Find the greatest common factor of the monomials.
10. 16xy 2, 12x 2y
11. 20m, 15m 3n
12. 4pq 4, 8p 3q 2
4xy
5m
4pq 2
Lesson 4.2
Greatest Common Factor
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EXAMPLE
4
Using the Greatest Common Factor
Party Favors You are making favors for a party. You buy 84 pencils,
42 gift certificates, 126 stickers, and 168 pieces of candy. Every favor
must have the same contents, and there should be no leftover items.
What is the greatest number of favors that you can make?
Solution
You can find the greatest number of favors by finding the GCF.
84 22 p 3 p 7
42 2 p 3 p 7
126 2 p 32
168 23
p7
p3p 7
The common prime factors are 2 , 3 , and 7 . The GCF is
2 p 3 p 7 , or 42 .
ANSWER The greatest number of favors is 42 . Each favor contains
2 pencils, 1 gift certificate, 3 stickers, and 4 pieces of candy.
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