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5-3 Least Common Multiple
Name
Date
Here are two ways to find the LCM of the
numbers 9, 12, and 18:
Method 1 List the multiples.
Multiples of 9 ⫽ 9, 18, 27, 36, . . .
Multiples of 12 ⫽ 12, 24, 36, . . .
Multiples of 18 ⫽ 18, 36, . . .
The LCM of 9, 12, and 18 is 36.
Find the least common denominator (LCD)
7
of 56 , 12
, and 38 .
List the multiples of each denominator.
Then find the first common multiple, or LCM.
6: 6, 12, 18, 24, . . .
Method 2 Use prime factorization.
9⫽3•3
12 ⫽ 2 • 2 • 3
18 ⫽ 2 • 3 • 3
Write the greatest number of times each
factor occurs among all the numbers.
The product of the factors is the LCM.
The LCM is 2 • 2 • 3 • 3 ⫽ 36.
12: 12, 24, . . .
8: 8, 16, 24, . . .
The LCM of 6, 8, and 12 is 24.
7
, and 38 is 24.
So the LCD of 56 , 12
You can also use the
prime factorization of
each denominator to
find the LCD in the
same way that you
found the LCM.
Two numbers are relatively
prime when their only
common factor is 1. The
LCM of relatively prime
numbers is their product.
Find the LCM for each set of numbers. List the multiples.
1. 6 and 10
Multiples of
6 ⴝ 6, 12, 18, 24, 30
Multiples of
10 ⴝ 10, 20, 30
Copyright © by William H. Sadlier, Inc. All rights reserved.
LCM: 30
5. 3, 2, and 8
2. 9 and 12
3. 2 and 5
4. 9 and 4
Multiples of
9 ⴝ 9, 18, 27, 36
Multiples of
12 ⴝ 12, 24, 36
Multiples of
2 ⴝ 2, 4, 6, 8, 10
Multiples of 5 ⴝ 5, 10
Multiples of
9 ⴝ 9, 18, 27, 36
Multiples of 4 ⴝ 4, 8,
12, 16, 20, 24, 32, 36
LCM: 36
LCM: 10
LCM: 36
6. 2, 4, and 6
7. 6, 8, and 4
8. 6, 8, and 9
Multiples of 3 ⴝ 3, 6,
9, 12, 15, 18, 21, 24
Multiples of 2 ⴝ 2, 4, 6, 8,
10, 12, 14, 16, 18, 20, 22, 24
Multiples of 8 ⴝ 8, 16, 24
Multiples of 2 ⴝ 2,
4, 6, 8, 10, 12
Multiples of 4 ⴝ 4, 8, 12
Multiples of 6 ⴝ 6, 12
Multiples of
6 ⴝ 6, 12, 18, 24
Multiples of 8 ⴝ 8, 16, 24
Multiples of 4 ⴝ 4,
8, 12, 16, 20, 24
Multiples of 6 ⴝ 6,
12, 18, 24, 30, 36, 42,
48, 54, 60, 66, 72
Multiples of 8 ⴝ 8, 16,
24, 32, 40, 48, 56, 64, 72
Multiples of 9 ⴝ 9, 18,
27, 36, 45, 54, 63, 72
LCM: 24
LCM: 12
LCM: 24
LCM: 72
Find the LCM for each set of numbers. Use prime factorization.
9. 6 and 9
LCM: 18
13. 4, 6, 3, and 9
LCM: 36
10. 8 and 12
11. 9 and 18
LCM: 24
LCM: 18
14. 9, 12, 6, and 18
15. 4, 7, 8, and 14
LCM: 36
LCM: 56
Lesson 5-3, pages 112–113.
12. 48 and 12
LCM: 48
16. 8, 16, 12, and 48
LCM: 48
Chapter 5
127
For More Practice Go To:
Use number sense to find the LCD. Describe the pattern in each row.
3
17. 10
13 and 5
4
18. 37 and 11
LCD: 65
19. 15 and 12
LCD: 77
23. 12 , 38 , and 34
LCD: 12
LCD: 21
use the product of the prime number denominators.
21. The LCD pattern above is to
5
22. 34 , 12
, and 23
LCD: 10
20. 37 and 23
1
9
24. 27 , 14
, and 28
LCD: 8
LCD: 28
8 7
1
25. 15
, 60 , and 12
LCD: 60
26. The LCD pattern above is to use the greatest denominator, since the other denominators are factors of it.
Find the LCD. List the multiples or use prime factorization.
1
2
28. 56 , 12
, and 11
LCD: 88
LCD: 132
5 25
31. 18
, 27, and 25
2
32. 16 , 17
, and 10
51
LCD: 270
LCD: 140
7 5
33. 20
, 10, and 13
15
LCD: 102
35. Mr. Gupta drew this diagram with prime
factors for 30 and 70. Can he use the diagram
to find the GCF, and LCM? Explain.
30
3
7 13
2
29. 20
, 35, and 14
70
2
5
7
Yes, the overlap indicating 2 • 5 shows the
GCF (10). The entire Venn diagram shows
the LCM, which is the product of all the
numbers: 3 • 2 • 5 • 7 ⴝ 210.
LCD: 60
3
5
30. 18
18, 27, and 45
LCD: 9
7 3
34. 21
, 9, and 57
LCD: 63
36. Louise wants to make cheeseburgers for
a company picnic. The hamburger patties
come in packs of 24, 18 buns are in a bag,
and there are 30 cheese slices per pack. If
Louise wants no leftovers, how many packs
of each item will she need?
18 ⴝ 2 • 32, 24 ⴝ 23 • 3, 30 ⴝ 2 • 3 • 5
LCM ⴝ 23 • 32 • 5 ⴝ 360
patties: 360 ⴜ 24 ⴝ 15, buns: 360 ⴜ 18 ⴝ 20,
cheese: 360 ⴜ 30 ⴝ 12
Louise needs 15 packs of patties, 20 packs
of buns, and 12 packs of cheese slices.
37. A pair of numbers has a GCF of 6 and a LCM of 60.
What could the numbers be? Explain.
Possible response: The numbers could be 12 and 30. The GCF indicates that each number is a
multiple of 6 and has 2 and 3 as common factors. Factoring the LCM shows 60 ⴝ 5 • 3 • 2 • 2. By
guess and test, the numbers could be 30 and 12; each are multiples of 6, and have a LCM of 60.
128
Chapter 5
Copyright © by William H. Sadlier, Inc. All rights reserved.
4 21
, 22 and 53
27. 11
88