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LESSON
4.2
Greatest Common Factor
BEFORE
Vocabulary
common factor, p. 179
greatest common
factor (GCF), p. 179
relatively prime, p. 180
Now
WHY?
You found all the factors You’ll find the GCF of two or
of a whole number.
more whole numbers.
So you can organize bands at a
music camp, as in Ex. 32.
Music Choir A choir director wants
to divide a choir into smaller groups.
The choir has 24 sopranos, 60 altos,
and 36 tenors. Each group will have
the same number of each type of
voice. What is the greatest number
of groups that can be formed? How
many sopranos, altos, and tenors will
be in each group?
A common factor is a whole number that is a factor of two or more
nonzero whole numbers. The greatest of the common factors is the
greatest common factor (GCF) .
Example 1
Finding the Greatest Common Factor
For the choir described above, the greatest number of groups that
can be formed is given by the GCF of 24, 60, and 36. You can use one
of two methods to find the GCF.
Method 1 List the factors of each number. Identify the greatest
number that is on every list.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12 , 15, 20, 30, 60
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.
Method 2 Write the prime factorization of each number. The GCF is
Watch Out
In Method 2 of Example 1,
the number 2 appears at least
twice in the prime factorization
of each number. So, include 2
twice when finding the GCF.
the product of the common prime factors.
24 2 p 2 p 2 p 3
The common prime factors
are 2, 2, and 3. The GCF is
60 2 p 2 p 3 p 5
the product 2 p 2 p 3 ⴝ 12.
36 2 p 2 p 3 p 3
Answer The greatest number of groups that can be formed is 12.
Each group will have 24 12 2 sopranos, 60 12 5 altos, and
36 12 3 tenors.
Checkpoint
Find the greatest common factor of the numbers.
1. 12, 30
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2. 21, 42
3. 16, 32, 40
Lesson 4.2
4. 27, 45, 90
Greatest Common Factor
179
Relatively Prime Two or more numbers are relatively prime if their
greatest common factor is 1.
Example 2
Note Worthy
To clarify the meaning of a
vocabulary term like relatively
prime, include both examples
and nonexamples of the term
in your notebook.
Identifying Relatively Prime Numbers
Find the greatest common factor of the numbers. Then tell
whether the numbers are relatively prime.
a. 24, 45
b. 35, 54
Solution
a. List the factors of each number. Identify the greatest number
that the lists have in common.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 45: 1, 3 , 5, 9, 15, 45
The GCF is 3. So, the numbers are not relatively prime.
b. Write the prime factorization of each number.
35 5 p 7
54 2 p 3 p 3 p 3
There are no common prime factors. However, two numbers
always have 1 as a common factor. So, the GCF is 1, and the
numbers are relatively prime.
Checkpoint
Find the greatest common factor of the numbers. Then tell whether
the numbers are relatively prime.
5. 18, 33
6. 39, 50
7. 110, 77
8. 21, 160
9. Critical Thinking Suppose you divide two numbers by their greatest
common factor. What is the relationship between the resulting
quotients?
Monomials and the GCF You can find the greatest common factor of two
or more monomials by factoring each monomial.
Example 3
Finding the GCF of Monomials
Find the greatest common factor of 18xy 2 and 28x 2y 2.
Factor the monomials. The GCF is the product of the common factors.
18xy 2 2 p 3 p 3 p x p y p y
28x 2y 2 2 p 2 p 7 p x p x p y p y
Answer The GCF is 2xy 2.
Checkpoint
Find the greatest common factor of the monomials.
10. 6x, 15x
180
Chapter 4
Factors, Fractions, and Exponents
11. 20x 2, 36x
12. 32y 2, 6x 2y
13. 7xy 3, 28xy 2
4.2
Exercises
More Practice, p. 842
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Practice Exercises
Guided Practice
Vocabulary Check
1. What does it mean for a number to be a common factor of two
numbers?
2. Find two pairs of relatively prime numbers from 5, 10, 16, and 25.
Skill Check
Find the greatest common factor of the numbers. Then tell whether the
numbers are relatively prime.
3. 7, 28
4. 34, 38
5. 11, 51
6. 32, 81
Find the greatest common factor of the monomials.
7. 18c, 4c
Guided Problem Solving
8. r, r 4
9. 5m, 20m3
10. 3x 2, 15x 3
11. Art Supplies To celebrate a grand opening,
the owner of an art supplies store is making
free gift bags for customers. The owner has
225 pastel crayons, 75 paintbrushes, and
120 tubes of oil paint. Each gift bag must be
identical. What is the greatest number of gift
bags the owner can make?
1
Write the prime factorization of
each number.
2
What are the common prime factors
of the numbers? What is the GCF
of the numbers?
3
What does the GCF represent in this
situation?
Practice and Problem Solving
Homework Help
Example
1
2
3
Exercises
12–19, 32–33
20–27, 34–36
28–31, 37–45
Lesson Resources
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• More Examples
• @HomeTutor
Find the greatest common factor of the numbers.
12. 28, 42
13. 21, 99
14. 34, 85
15. 12, 36
16. 32, 55
17. 54, 89
18. 76, 86
19. 120, 960
Find the greatest common factor of the numbers. Then tell whether the
numbers are relatively prime.
20. 9, 26
21. 11, 55
22. 12, 33
23. 77, 51
24. 58, 60
25. 121, 280
26. 64, 144
27. 28, 84
Find the greatest common factor of the monomials.
28. 16x, 36x
29. 18m2, 7m
30. 18k, 15k 3
Lesson 4.2
31. 2x, 8x 2, 6x 3
Greatest Common Factor
181
32. Music Camp A summer music camp has 88 participants. The camp
has 32 vocalists, 16 drummers, 24 guitarists, and 16 bassists. What is
the greatest number of identical bands that can be formed using all the
participants? How many vocalists will be in each band?
33. Flower Bouquets The science
For help with reading bar
graphs, see p. 817.
Types of Flowers
Flowers
Review Help
club is selling flowers for a
fundraiser. The club wants to
make bouquets from 4 types of
flowers. The bar graph shows
how many flowers of each type
the club has. What is the greatest
number of identical bouquets
that can be made? What will each
bouquet contain?
70
56
42
28
14
0
Daisy
Lily
Iris
Freesia
Tell whether the numbers are relatively prime.
34. 115, 207
35. 224, 243
36. 152, 171
Find the greatest common factor of the monomials.
37. 12m2n3, 70m3n
38. 72a3b 2, 86a
39. 44m2n, 48mn2
40. a 2b 3, ab 3
41. 3x, 7xy 2
42. 4rs 2, 27st 3
43. 18wx 2, 45wx
44. 12y 2, 15y 3, 5y
45. rs3, s3t, r 2st 2
46. Community Garden You want to cover the walkway of a community
garden with square clay tiles. The space you want to cover is a
rectangle 42 inches wide by 72 inches long. Assuming you want to
cover the space exactly without cutting any tiles, what is the greatest
side length you can use for the tiles?
47. Bracelets You want to make woven plastic bracelets. You have
3 pieces of plastic lacing with lengths 45 cm, 75 cm, and 60 cm. You
need to cut the lacing into pieces of the same length. What is the
greatest possible length each piece can be, without any lacing being
wasted?
48. Critical Thinking The greatest common factor of 30 and a
number n is 6. Find a possible value for n. Are there other possible
values for n? Explain.
49. Extended Problem Solving In the future, scientists may want to
In the
Real World
Mars Shown above is an
illustration of what a base on
Mars might look like. On Mars,
there are about 669 solar days
in one year. Assuming each day
has 1480 minutes, about how
many minutes are there in one
year on Mars?
make a unit of time that is convenient for people living on both Earth
and Mars. The new unit of time, called the space-hour, should divide
evenly into the number of minutes in each planet’s day. Under the
current Earth definition of minutes, Earth has 1440 minutes per day,
and Mars has approximately 1480 minutes per day.
a. Analyze What is the greatest number of minutes that could be in a
space-hour?
b. Apply How many space-hours would there be each day on Earth? on
Mars?
c.
182
Chapter 4
A spacecraft that uses current technology can take 210 days to
travel from Earth to Mars. Use a calculator to find how long this
trip would be in space-hours.
Factors, Fractions, and Exponents
50.
Writing If a and b are nonzero whole numbers and a is a factor of
b, what is the GCF of a and b? Explain your thinking and give three
numerical examples to support your answer.
51. Critical Thinking If a and b are relatively prime numbers and b and c
are relatively prime numbers, are a and c relatively prime numbers?
Give examples to support your answer.
52. Challenge Consider the pattern 2x, 6x 2, 18x 3, 54x 4, . . . . What are
the next two monomials in the pattern? What is the GCF of all the
monomials in the pattern? What is the GCF of all the monomials in
the pattern excluding the first monomial?
Mixed Review
Find the sum or difference. (p. 815)
5
2
53. _ _
9
9
3
3
54. _ _
7
7
8
14
55. _ _
3
11
56. _ _
5
59. _ 36
3
60. _ 49
15
15
20
20
Find the product. (p. 816)
3
57. 60 _
10
1
58. 28 _
4
12
7
Write the prime factorization of the number. (Lesson 4.1)
61. 125
Standardized Test
Practice
62. 70
63. 52
64. 200
65. Multiple Choice Which numbers are not relatively prime?
A. 32, 65
B. 34, 69
C. 63, 91
D. 26, 85
66. Short Response You are making first-aid kits to go camping. You have
48 bandages, 15 squares of gauze, 6 tubes of antibiotic ointment, and
6 ice packs. What is the greatest number of identical first-aid kits that
you can make? How many of each item will each first-aid kit contain?
Common Factor Commotion
Each number in the third column of the table is the greatest
common factor of the numbers in the same row. Each number
in the first two columns has exactly one digit that is different
from the number above it and exactly one digit that is different
from the number below it.
Copy the table and fill in
each of the blanks with a
number that satisfies the
conditions.
First number
Second number
GCF
945
?
648
?
?
435
432
532
105
15
?
14
Lesson 4.2
Greatest Common Factor
183