4 . Least Common Multiple Now

LESSON
Page 1 of 5
4.4
Least Common Multiple
Now
BEFORE
WHY?
Vocabulary
You found the GCF of
two numbers.
You’ll find the least common
multiple of two numbers.
So you can design a fitness
schedule, as in Ex. 38.
multiple, p. 187
common multiple,
p. 187
least common multiple
(LCM), p. 187
least common
denominator (LCD),
p. 188
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.
Example 1
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.
62 p 3
10 2 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
Lesson 4.4
4. 15, 30, 50
Least Common Multiple
187
Page 2 of 5
Example 2
Finding the Least Common Multiple of Monomials
Find the least common multiple of 9xy 2 and 15x 2y.
9xy 2 3 p 3 p x p y p y
15x 2y 3 p 5 p x p x p y
LCM 3 p x p y p 3 p 5 p x p y 45x 2y 2
Study Strategy
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.
Common factors are
circled and used only
once in the LCM.
Answer The least common multiple of 9xy 2 and 15x 2y is 45x 2y 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.
Example 3
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.
Number of snowboarders
Total number of visitors
45,000
144,000
5
16
Number of snowboarders
Total number of visitors
56,000
160,000
7
20
Last year: This year: 2
5
16
7
20
Find the LCD of and .
The LCM of 16 and 20 is 80. So, the LCD of the fractions is 80.
3
Write equivalent fractions using the LCD.
5
16
5p5
16 p 5
25
80
7
20
Last year: 4
25
80
7p4
20 p 4
28
80
This year: 28
80
5
16
7
20
Compare the numerators: < , so < .
Answer The fraction of snowboarders was greater this year.
Checkpoint
Find the least common multiple of the monomials.
5. 15x 2, 27x
6. 6m2, 10m3
7. 14ab, 21bc
8. r 2, 5rst
Use the LCD to determine which fraction is greater.
5 7
9. , 6 9
188
Chapter 4
Factors, Fractions, and Exponents
5 13
10. , 8 20
7 11
11. , 12 15
5 3
12. , 16 10
Page 3 of 5
Example 4
Ordering Fractions and Mixed Numbers
4 33
1 5 10
19
6
Order the numbers 3 , , and from least to greatest.
Review Help
1
Write the mixed number as an improper fraction.
4
15
3 p 15 4
15
49
15
3 For help with writing mixed
numbers as improper
fractions, see p. 778.
2
49 33
15 10
19
6
Find the LCD of , , and .
The LCM of 15, 10, and 6 is 30. So, the LCD is 30.
3
Write equivalent fractions using the LCD.
49
49 p 2
98
15
15 p 2
30
4
33
33 p 3
99
10
10 p 3
30
19
19 p 5
95
6
6p5
30
Compare the numerators.
95 98
98 99
19 49
49 33
< and < , so < and < .
30 30
30 30
6
15
15 10
19
6
4
15
33
10
Answer From least to greatest, the numbers are , 3 , and .
4.4
Exercises
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Guided Practice
Vocabulary Check
1. How are the terms least common multiple and least common
denominator related?
4
7
2. Describe how you would use the LCD to compare and .
7
12
Skill Check
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. 3s, s 2
8. x 4, x 2
9. 15m 2, 9m
10. 8b, 20b 2
Use the LCD to determine which fraction is greater.
3 5
11. , 4 8
2 13
12. , 3 16
3 7
14. , 4 10
2 3
13. , 5 8
15. Error Analysis Describe
16 24
and correct the error in
finding the LCM of 16
and 30.
30 2 p 3 p 5
LCM 2 p 3 p 5 480
5
Lesson 4.4
Least Common Multiple
189
Page 4 of 5
Practice and Problem Solving
Homework Help
Example
1
2
3
4
Exercises
16–27, 36, 38
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
19. 10, 15
20. 21, 14
21. 30, 36
22. 55, 15
23. 42, 66
24. 3, 6, 12
25. 8, 11, 36
26. 10, 12, 14
27. 16, 20, 30
Find the least common multiple of the monomials.
28. 5a 2, 16a 3
29. 21w, 9w 2
30. 17b 2, 3b 3
31. 14x 4, 21x 2
32. 60s4, 24s 3
33. 2n3, 8n2
34. 25a, 40a 2
35. 11s, 33s 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.
Writing
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.
1 2
39. , 4 7
2 5
40. , 3 8
7 11
41. , 10 15
3 6
42. , 5 11
5 4
43. , 12 15
7 9
44. , 20 25
5 8
45. , 18 21
11 20
46. , 42 63
Order the numbers from least to greatest.
7 11 1
47. , , 1 6 9
3
13 1 27
48. , 3 , 4
2 8
8 1 3
49. , , 15 5 10
5 14 9
50. , , 11 33 22
3 4 7
51. , , 4 9 15
5 7 11
52. , , 6 10 15
5 43
12
53. , 2 , 12 18
5
1 10 13
54. 1, , 1
3 7
33
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. 24de 2, 36d 3e
57. x 3y, 15xy 5
58. 10a 2b 2, 20ab 59. 45gh3, 33g 4h
60. xyz 3, x 2yz 2
61. 26ab2, 28ac3
62. 11rst, 15r 3t 2
63. 30df 2, 40d 3ef
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
2a
5b
3
4ab
To rewrite and 2 with a common denominator, first find the
LCD of the fractions.
The LCM of 5b and 4ab 2 is 20ab 2. So, the LCD is 20ab 2.
Then write equivalent fractions using the LCD.
2a
2a p 4ab
8a 2b
2
5b
5b p 4ab
20ab
3
3p5
15
2 2 2
4ab
4ab p 5
20ab
x y
66. , 6 y 8x
x x
65. , 3 4
3x 2
67. 2 , 4y 5xy
3x 5y
68. , 2yz 4xz
69. Critical Thinking Let a and b represent nonzero whole numbers.
a
1 a a 2
Find a fraction such that < , < , and b < 30.
b
6 b b 7
70. Challenge 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.
Prime factorizations LCM GCF LCM p GCF a p b
Given numbers
Mixed Review
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. n2
72. n3
73. n4
74. n5
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)
Standardized Test
Practice
80. Multiple Choice Which expression is the least common multiple of the
monomials 27w 4z and 75w 2z 2?
A. 3w 2z
B. 75w 4z 2
C. 675w 2z
D. 675w 4z 2
81. Multiple Choice Which list shows the fractions in order from least to
greatest?
2 1 4
F. 9, 6, 25
3 11 9
G. 7, 24, 21
7 3 5
H. 20, 8, 12
Lesson 4.4
2 19 21
I. 5, 40, 45
Least Common Multiple
191