Page 1 of 5
4 .
4
multiple, p. 187 common multiple, p. 187 least common multiple
(LCM), p. 187 least common denominator (LCD), p. 188
B E F O R E
You found the GCF of two numbers.
Now
You’ll find the least common multiple of two numbers.
W H Y ?
So you can design a fitness schedule, as in Ex. 38.
Agriculture Crop rotation is a system in which farmers vary the crops they plant in their fields each year. Suppose a farmer grows alfalfa in a certain field every 6 years. In another field, the farmer grows alfalfa every
10 years. This year, the farmer is growing alfalfa in both fields. In how many years will the farmer grow alfalfa in both fields again?
A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple . Some of the common multiples of
8 and 12 are shown in blue below.
Multiples of 8: 8, 16, 24 , 32, 40, 48 , 56, 64, 72 , 80, . . .
Multiples of 12: 12, 24 , 36, 48 , 60, 72 , 84, 96, . . .
The least of the common multiples of two or more numbers is the least common multiple (LCM) . The LCM of 8 and 12 is 24.
Finding the Least Common Multiple
For the crop rotation system described above, the number of years until the farmer grows alfalfa in both fields again is given by the LCM of 6 and 10. You can use one of two methods to find the LCM.
Method 1 List the multiples of each number. Identify the least number that is on both lists.
Multiples of 6: 6, 12, 18, 24, 30 , 36, 42, 48, 54, 60
Multiples of 10: 10, 20, 30 , 40, 50, 60
The LCM of 6 and 10 is 30.
Method 2 Find the common factors of the numbers.
6 2
10 2 p
3 p 5
The common factor is 2.
Multiply all of the factors, using each common factor only once.
LCM 2 p 3 p 5 30
Answer The farmer will grow alfalfa in both fields again in 30 years.
Checkpoint
Find the least common multiple of the numbers.
1.
16, 24 2.
20, 25 3.
6, 8, 20 4.
15, 30, 50
Lesson 4.4
Least Common Multiple 187

Page 2 of 5
In Example 2, notice that the
LCM of the two monomials includes the higher power of each variable, as well as the higher power of each prime number factor.
Finding the Least Common Multiple of Monomials
Find the least common multiple of 9 xy
2 and 15 x
2 y .
9 xy
2
3 p
3 p x p y p
15 x
2 y 3 p 5 p x p x p y y
LCM 3 p x p y p 3 p 5 p x p y 45 x
2 y
2
Common factors are circled and used only once in the LCM.
Answer The least common multiple of 9 xy
2 and 15 x
2 y is 45 x
2 y
2
.
Least Common Denominator The least common denominator (LCD) of two or more fractions is the least common multiple of the denominators.
You can use the LCD to compare and order fractions.
Comparing Fractions Using the LCD
Winter Sports Last year, a winter resort had 144,000 visitors, including
45,000 snowboarders. This year, the resort had 160,000 visitors, including 56,000 snowboarders. In which year was the fraction of snowboarders greater?
Solution
1 Write the fractions and simplify.
Last year :
Number of snowboarders
Total number of visitors
This year :
Number of snowboarders
Total number of visitors
4 5 ,0 0 0 5
1 4 4 ,0 0 0 1 6
5 6 ,0 0 0 7
1 6 0 ,0 0 0 2 0
2 Find the LCD of
5
1 6 and
7
2 0
.
The LCM of 16 and 20 is 80. So, the LCD of the fractions is 80.
3 Write equivalent fractions using the LCD.
Last year :
5 5
1 6 1 6 p p
5
5
2 5
8 0
This year :
7 7
2 0 2 0 p p
4
4
4 Compare the numerators:
2 5
8 0
<
2 8
8 0
, so
5
1 6
<
7
2 0
.
Answer The fraction of snowboarders was greater this year.
2 8
8 0
Checkpoint
Find the least common multiple of the monomials.
5.
15 x
2
, 27 x 6.
6 m
2
, 10 m
3
7.
14 ab , 21 bc
Use the LCD to determine which fraction is greater.
9.
5
6
,
7
9
10.
5
8
,
1 3
2 0
11.
7
1 2
,
1 1
1 5
8.
r
2
, 5 rst
12.
5
1 6
,
3
1 0
188 Chapter 4 Factors, Fractions, and Exponents

Page 3 of 5
Help
For help with writing mixed numbers as improper fractions, see p. 778.
Ordering Fractions and Mixed Numbers
Order the numbers 3
4
1 5
,
3 3
, and
1 0
1 9
6 from least to greatest.
1 Write the mixed number as an improper fraction.
3
4
1 5
3 p
1 5
1 5
4 4 9
1 5
2
3
Find the LCD of
4 9
,
1 5
3 3
, and
1 0
1 9
.
6
The LCM of 15, 10, and 6 is 30. So, the LCD is 30.
Write equivalent fractions using the LCD.
4 9
1 5
4 9
1 5 p p
2
2
9 8
3 0
3 3
1 0
3 3
1 0 p p
3
3
9 9
3 0
1 9 1
6 6
9 p p
5
5 9 5
3 0
4 Compare the numerators.
9 5
3 0
< 9 8
3 0 and
9 8
3 0
< 9 9
3 0
, so
1 9
6
< 4 9
1 5 and
4 9
1 5
< 3 3
1 0
.
Answer From least to greatest, the numbers are
1 9
6
, 3
4
1 5
, and
3 3
1 0
.
4 .
4
INTERNET eWorkbook Plus
CLASSZONE.COM
1.
How are the terms least common multiple and least common denominator related?
2.
Describe how you would use the LCD to compare
4
7 and
7
1 2
.
Find the least common multiple of the numbers.
3.
3, 4 4.
4, 8 5.
18, 24 6.
10, 16
Find the least common multiple of the monomials.
7.
3 s , s
2
8.
x
4
, x
2
9.
15 m
2
, 9 m 10.
8 b , 20 b
2
Use the LCD to determine which fraction is greater.
11.
3
4
,
5
8
12.
2
3
,
1 3
1 6
13.
2
5
,
3
8
15.
Describe and correct the error in finding the LCM of 16 and 30.
14.
3
4
,
7
1 0
16 2
4
30 2 p
3 p
5
LCM 2
5 p 3 p 5 480
Lesson 4.4
Least Common Multiple 189

Page 4 of 5
Help
Example Exercises
1 16–27, 36, 38
2
3
4
28–35
39–46
47–54
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
Find the least common multiple of the numbers.
16.
9, 12 17.
3, 8 18.
4, 16
20.
21, 14
24.
3, 6, 12
21.
25.
30, 36
8, 11, 36
22.
26.
55, 15
10, 12, 14
19.
10, 15
23.
42, 66
27.
16, 20, 30
Find the least common multiple of the monomials.
28.
5 a
2
, 16 a
32.
60 s
4
, 24 s
3
3
29.
21 w
33.
2 n
3
, 9 w
, 8 n
2
2
30.
17 b
2
, 3 b
3
34.
25 a , 40 a
2
31.
14 x
4
, 21 x
2
35.
11 s , 33 s
2
36.
Visual Patterns In the first pattern shown below, the green star repeats every 6 figures. In the second pattern, the green star repeats every
8 figures. How many figures after the first figure will both patterns have a green star?
37.
Could you find the greatest common multiple of two numbers? Explain your thinking.
38.
Fitness You lift weights every third day and take karate class every
Monday. Today you have karate and are lifting weights. In how many days will you next lift weights and have karate on the same day?
Use the LCD to determine which fraction is greater.
39.
43.
1
4
,
2
7
5
1 2
,
4
1 5
40.
44.
2
3
,
5
8
7
2 0
,
9
2 5
41.
7
1 0
,
1 1
1 5
45.
5
1 8
,
8
2 1
42.
3
5
,
6
1 1
46.
1 1
4 2
,
2 0
6 3
Order the numbers from least to greatest.
47.
7
6
,
1 1
9
, 1
1
3
48.
1 3
4
, 3
1
2
,
2 7
8
49.
8
1 5
,
1
5
,
3
1 0
50.
5
1 1
,
1 4
3 3
,
9
2 2
51.
3
4
,
4
9
,
7
1 5
52.
5
6
,
7
1 0
,
1 1
1 5
53.
1 2
, 2
5
5
1 2
,
4 3
1 8
54.
1
1
3
,
1 0
, 1
7
1 3
3 3
55.
Critical Thinking What is the least number for which the LCM of the number and 12 is 300? Explain your thinking.
Find the least common multiple of the monomials.
56.
24 de
2
, 36 d
3 e 57.
x
3 y , 15 xy
5
58.
10 a
2 b
2
, 20 ab 59.
45 gh
3
, 33 g
4 h
60.
xyz
3
, x
2 yz
2
61.
26 ab
2
, 28 ac
3
62.
11 rst , 15 r
3 t
2
63.
30 df
2
, 40 d
3 ef
64.
Vice Presidents During the period 1800–1900, 6 out of 23 U.S. Vice
Presidents later became U.S. Presidents. During the period 1901–2000,
7 out of 21 Vice Presidents later became Presidents. During which period did a greater fraction of Vice Presidents become Presidents?
190 Chapter 4 Factors, Fractions, and Exponents

Page 5 of 5
In Exercises 65–68, rewrite the variable expressions with a common denominator.
Rewriting Variable Expressions
To rewrite
2 a
5 b and
3
4 a b
2
LCD of the fractions.
with a common denominator, first find the
The LCM of 5 b and 4 ab
2 is 20 ab
2
. So, the LCD is 20 ab
2
.
Then write equivalent fractions using the LCD.
2 a
5 b
2 a
5 b p p
4 a b
4 a b
8 a
2 b
2 0 a b
2
3
4 a b
2
3 p
4 a b
2
5 p
1 5
5 20 a b
2
65.
x
3
, x
4
66.
x
6 y
, y
8 x
67.
3 x
4 y
2
,
2
5 xy
68.
3 x
2 y z
,
5 y
4 x z
69.
Critical Thinking Let a and b represent nonzero whole numbers.
Find a fraction a b such that
1
6
< a b
, a b
< 2
7
, and b
<
30.
70.
Copy and complete the table for the given values of a and b . Describe any relationships you notice between the product of the
LCM and the GCF and the product of a and b .
Given numbers Prime factorizations LCM GCF LCM p
GCF a p b a 6, b 18 a 15, b 35 a 6, b 20 a 12, b 60
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Evaluate the expression when n 5.
(Lesson 1.2)
71.
n
2
72.
n
3
73.
n
4
74.
n
5
Write the prime factorization of the number.
(Lesson 4.1)
75.
28 76.
39 77.
81 78.
165
79.
Cookies You are making gift boxes filled with cookies to give to friends.
You have 64 peanut butter cookies, 80 chocolate chip cookies, and
56 sugar cookies. What is the greatest number of identical gift boxes that you can make?
(Lesson 4.2)
80.
Multiple Choice Which expression is the least common multiple of the monomials 27 w
4 z and 75 w
2 z
2
?
A.
3 w
2 z B.
75 w
4 z
2
C.
675 w
2 z D.
675 w
4 z
2
81.
Multiple Choice Which list shows the fractions in order from least to greatest?
F.
2
9
,
1
6
,
4
2 5
G.
3
7
,
1 1
,
2 4
9
2 1
H.
7
2 0
,
3
8
,
5
1 2
I.
2
5
,
1 9
,
4 0
2 1
4 5
Lesson 4.4
Least Common Multiple 191