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Greatest Common Factor
BEFORE
Now
WHY?
You found all the factors
of a whole number.
You’ll find the greatest common
factor of two or more numbers.
So you can decorate Rose Bowl
floats, as in Ex. 38.
In the Real World
Word Watch
common factor, p. 164
greatest common factor
(GCF), p. 164
relatively prime, p. 165
Orchestra An orchestra conductor
divides 48 violinists, 24 violists,
and 36 cellists into ensembles.
Each ensemble has the same
number of each instrument.
What is the greatest number of
ensembles that can be formed?
How many violinists, violists, and
cellists will be in each ensemble?
A whole number that is a factor of two or more nonzero whole numbers
is called a common factor . The greatest of the common factors is called
the greatest common factor (GCF) . One way to find the greatest
common factor of two or more numbers is to make a list of all the factors
of each number and identify the greatest number that is on every list.
EXAMPLE
1
Making a List to Find the GCF
In the orchestra problem above, the greatest number of ensembles that
can be formed is given by the greatest common factor of 48, 24, and 36.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The common factors
are 1, 2, 3, 4, 6, and
12. The GCF is 12.
ANSWER The greatest common factor of 48, 24, and 36 is 12. So, the
greatest number of ensembles that can be formed is 12. Then each
ensemble will have 4 violinists, 2 violists, and 3 cellists.
Your turn now
Find the greatest common factor of the numbers by
listing factors.
164
Chapter 4
1. 16, 28
2. 21, 42
3. 60, 96
4. 12, 33, 39
5. 14, 35, 63
6. 32, 40, 64
Number Patterns and Fractions
Page 2 of 4
Using Prime Factorization Another way to find the greatest common
factor of two or more numbers is to use the prime factorization of each
number. The product of the common prime factors is the greatest
common factor.
EXAMPLE
with
Solving
Large numbers may have
many factors, and it may
be difficult to list all the
factors. Sometimes it’s
easier to use prime
factorization to find the
greatest common factor of
large numbers.
2
Using Prime Factorization to Find the GCF
Find the greatest common factor of 180 and 126 using prime
factorization.
Begin by writing the prime factorization of each number.
180
126
10 18
2 63
2 5 2 9
2 3 21
2 5 2 3 3
2 3 3 7
180 2 2 3 3 5
126 2 3 3
7
ANSWER The common prime factors of 180 and 126 are 2, 3, and 3.
So, the greatest common factor is 2 32 18.
Your turn now
Find the greatest common factor of the numbers using
prime factorization.
7. 90, 150
8. 84, 216
9. 120, 192
10. 105, 225
Relatively Prime Two or more numbers are relatively prime if their
greatest common factor is 1.
tch Out!
Wa
EXAMPLE
3
Identifying Relatively Prime Numbers
Tell whether the numbers are relatively prime.
To say that two
numbers are relatively
prime does not mean
that one of the numbers
is prime.
a. 28, 45
Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 45: 1, 3, 5, 9, 15, 45
The GCF is 1.
ANSWER Because the GCF is 1, 28 and 45 are relatively prime.
b. 15, 51
Factors of 15: 1, 3, 5, 15
The GCF is 3.
Factors of 51: 1, 3, 17, 51
ANSWER Because the GCF is 3, 15 and 51 are not relatively prime.
Lesson 4.2
Greatest Common Factor
165
Page 3 of 4
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 708
Getting Ready to Practice
1. Vocabulary Copy and complete: The numbers 35 and 36 are _?_
because their _?_ is 1.
Find the greatest common factor of the numbers by listing factors.
2. 14, 21
3. 24, 32
4. 20, 55, 65
5. 42, 72, 84
Find the greatest common factor of the numbers using prime
factorization. Then tell whether the numbers are relatively prime.
6. 98, 140
7. 27, 117
8. 56, 88
9. 72, 169
10. Science Class A science class with 15 girls and 12 boys is divided into
groups where each group has the same number of boys and the same
number of girls. What is the greatest number of groups that can be
formed? How many boys and girls are in each group?
Practice and Problem Solving
with
Example
1
2
3
Homework
Exercises
11–18
20–28, 30–35, 38
20–27
Online Resources
Find the greatest common factor of the numbers by listing factors.
11. 56, 81
12. 39, 52
13. 24, 63
14. 45, 76
15. 75, 90, 105
16. 48, 64, 96
17. 18, 30, 60
18. 36, 54, 135
19. Writing In your own words, describe how to find the greatest common
factor of two numbers given their prime factorizations.
CLASSZONE.COM
• More Examples
• eTutorial Plus
Find the greatest common factor of the numbers using prime
factorization. Then tell whether the numbers are relatively prime.
20. 86, 154
21. 37, 93
22. 198, 216
23. 36, 168
24. 34, 85
25. 75, 285
26. 144, 264
27. 65, 112
28. Fruit Baskets A school is preparing
fruit baskets for a local nursing home.
There are 162 apples, 108 oranges,
and 180 bananas. If the baskets are
identical and there is no leftover
fruit, what is the greatest number
of baskets that can be made? How
many apples, oranges, and bananas
are in each basket?
166
Chapter 4
Number Patterns and Fractions
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Recreation
29. Critical Thinking The lesser of two numbers is a factor of the greater
number. What can you say about the GCF of the numbers?
Find the GCF of the numbers using prime factorization.
30. 63, 84, 126
31. 39, 65, 182
32. 110, 132, 176
33. 168, 210, 238
34. 70, 147, 175, 280
35. 68, 102, 136, 153
Tell whether the statement is always, sometimes, or never true.
36. The greatest common factor of two odd numbers is 2.
37. The greatest common factor of two even numbers is 2.
38. Rose Bowl Floats You are decorating a Rose Bowl float. There are
108 red roses, 144 white roses, 48 yellow roses, and 72 purple roses.
If bunches of roses are identical and there are no leftover roses, what
is the greatest number of bunches that can be made? How many roses
of each color are in each bunch?
39. Challenge The GCF of a number and 96 is 32. The sum of the number’s
digits is 13. Find two numbers that satisfy these conditions.
■
Rose Bowl Floats
Every square inch of a
Rose Bowl float must be
covered by something
natural like flowers, seeds,
and leaves. Suppose it
takes 25 roses to cover
one square foot. How
many square feet do
20,000 roses cover?
Mixed Review
Find the quotient. Then check your answer. (Lesson 2.4)
40. 113.24 7.6
41. 27.44 1.4
42. 10.352 1.6
43. 15.67 2.5
Tell whether the number is prime or composite. (Lesson 4.1)
44. 41
45. 290
46. 57
47. 63
Basic Skills Solve the following problems.
48. A tube contains 18 lead refills for a mechanical pencil. If you buy
4 tubes, how many lead refills do you have?
49. You and a friend baked 54 brownies for a bake sale. The recipe says that
each batch yields 9 brownies. How many batches did you make?
Test-Taking Practice
INTERNET
50. Multiple Choice Identify which number pairs are relatively prime.
State Test Practice
I. 21, 32
II. 30, 36
III. 49, 72
CLASSZONE.COM
A. I and II
B. I and III
C. II and III
D. I, II, and III
51. Multiple Choice Find the greatest common factor of 180 and 225.
F. 9
G. 15
H. 25
Lesson 4.2
I. 45
Greatest Common Factor
167