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Least Common Multiple
BEFORE
Now
WHY?
You found greatest common
factors.
You’ll find least common
multiples.
So you can coordinate schedules,
as with the soccer games in Ex. 7.
In the Real World
Word Watch
multiple, p. 235
common multiple, p. 235
least common multiple
(LCM), p. 236
Ferry Boats Two ferry boats leave a
loading platform at the same time.
One of the ferry boats returns to the
loading platform every 25 minutes.
The other returns every 30 minutes.
In the next 300 minutes, when will
they return at the same time?
You can use multiples to answer the question above. A multiple of a
number is the product of the number and any nonzero whole number.
The three dots show that the
pattern continues forever.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, . . .
A multiple shared by two or more numbers is a common multiple .
EXAMPLE
1
Finding a Common Multiple
You can use common multiples to answer the question above about ferry
boats. Begin by writing the multiples of 25 and 30. Then identify
common multiples through 300.
Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300
The common multiples of 25 and 30 are 150 and 300.
ANSWER The ferry boats will return to the loading platform at the same
time in 150 minutes and in 300 minutes.
Your turn now
1. 2, 3
Find two common multiples of the numbers.
2. 3, 5
3. 8, 10
4. 6, 18
5. A cuckoo clock has birds that pop out of their nests every 6 minutes
and dancers that pop out every 15 minutes. Suppose that the birds
and dancers have just popped out at the same time. When will this
happen again in the next 60 minutes?
Lesson 5.4
Least Common Multiple
235
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Finding the Least Common Multiple (LCM)
The least common multiple of two or more numbers is the
smallest of the common multiples. Below are two methods to
find the LCM.
Method 1: Start listing the multiples of each number.
Then find the smallest of the common multiples.
Method 2: Write the prime factorizations of the numbers.
Multiply together the prime factors, using each prime
factor the greatest number of times it is a factor of any
of the numbers.
EXAMPLE
with
2
Finding the LCM
Find the LCM of 9 and 12.
Solving
If the only common factor
of two numbers is 1,
then their least common
multiple is the product
of the two numbers.
Multiples of 9: 9, 18, 27, 36, 45, 54, . . .
Multiples of 12: 12, 24, 36, 48, . . .
ANSWER The LCM of 9 and 12 is 36.
EXAMPLE
3
Using Prime Factorization
Find the LCM of 42 and 60 using prime factorization.
1 Write the prime factorizations. Circle any common factors.
42 2 3 7
60 2 2 3 5
2 Multiply together the prime factors, using each circled factor the
greatest number of times it occurs in either factorization.
2 2 3 5 7 420
ANSWER The LCM of 42 and 60 is 420.
236
Chapter 5
Your turn now
Find the LCM of the numbers.
6. 8, 12
7. 7, 8, 14
Number Patterns and Fractions
8. 50, 90
Page 3 of 4
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 712
Getting Ready to Practice
Vocabulary List the first three multiples of the number.
1. 7
2. 8
3. 11
4. 16
Use the prime factorizations to find the LCM of the numbers.
5. 32 2 2 2 2 2
6. 48 2 2 2 2 3
24 2 2 2 3
56 2 2 2 7
7. Guided Problem Solving You bring the drinks for your soccer team
every sixth game. Every third game is a home game. When will you
first bring the drinks to a home game? If there are 20 games in a
season, how many times will you bring the drinks to a home game
this season?
1 List the multiples of 6 and 3.
2 Find the common multiples of 6 and 3 from 1 to 20.
3 Use the results of Step 2 to answer the questions being asked.
Practice and Problem Solving
Find the LCM of the numbers by listing multiples.
8. 3, 7
12. 3, 6, 9
9. 5, 8
13. 4, 8, 16
16. Find the Error Describe
and correct the error
in finding the LCM
of 24 and 36.
with
Example
1
2
3
10. 6, 10
11. 10, 12
14. 2, 3, 5
15. 4, 9, 24
24 2 2 2 3
36 2 2 3 3
The LCM of 24 and 36 is 2 2 3 12.
Homework
Exercises
8–15, 25
8–15, 25
16–25
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
Find the LCM of the numbers using prime factorization.
17. 21, 28
18. 30, 42
19. 22, 36
20. 32, 40
21. 27, 45
22. 56, 64
23. 60, 72
24. 50, 75
25. Writing A store gives every 20th customer a $20 gift certificate. Every
75th customer gets a $75 gift certificate. Which customer will be the
first to receive both types of gift certificates? Explain how you found
your answer.
Lesson 5.4
Least Common Multiple
237
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26. Insects A cricket and a grasshopper are in a jumping contest.
The cricket jumps three inches every jump and the grasshopper
jumps four inches every jump. Name 4 points where the cricket
and the grasshopper will both land. How many jumps
will it take each of them to land at these points?
Number Sense Find a pair of numbers that matches the description.
27. The LCM of two prime numbers is 51.
28. The LCM of two numbers is 48. Their sum is 19.
29. The LCM of two numbers is 16. Their product is 64.
30. Challenge Find the GCF and the LCM of 6 and 12. How does the
product of the GCF and the LCM compare to the product of 6 and 12?
Try several pairs of numbers. What does this suggest about the product
of two whole numbers and the product of their GCF and LCM?
Mixed Review
31. Write two and fifty-six thousandths as a decimal. (Lesson 3.1)
Find the quotient. (Lesson 4.6)
32. 20.44 0.56
33. 13.08 0.24
34. 1.76 5.5
35. 1.8 7.2
Write two fractions that are equivalent to the given fraction.
(Lesson 5.3)
1
36. 8
3
37. 10
5
38. 12
12
39. 17
Basic Skills Copy and complete the statement with <, >, or .
40. 416 _?_ 419
41. 680 _?_ 68
42. 32 _?_ 352
43. 36 _?_ 390
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
44. Multiple Choice What is the least common multiple of 10 and 3?
A. 1
B. 13
C. 30
D. 60
45. Short Response Pencils come in packages of 10. Rulers come in
packages of 8. Hannah wants exactly one pencil for every ruler. What is
the smallest number of packages of each she will need to buy? Explain
how you found your answer.
238
Chapter 5
Number Patterns and Fractions