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Least Common Multiple
BEFORE
Now
WHY?
You found the GCF of
two or more numbers.
You’ll find the LCM of
two or more numbers.
So you can find when the Mayan
calendars coincide, as in Ex. 37.
In the Real World
Word Watch
multiple, p. 175
common multiple, p. 175
least common multiple
(LCM), p. 175
Model Trains You visit a model train
shop that has two working model
trains. The trains share a station,
but they run on separate tracks.
One of the trains returns to the
station every 4 minutes. The other
returns every 6 minutes. Both trains
just left the station. When will they
both return to the station?
A multiple of a 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 . The least of the common multiples is the
least common multiple (LCM) .
EXAMPLE
1
Using the Least Common Multiple
You can determine when the model trains described above will return to
the station by finding the least common multiple of 4 and 6. Begin by
writing the multiples of 4 and 6. Then identify any common multiples.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, . . .
Multiples of 6: 6, 12, 18, 24, 30, 36, . . .
12, 24, and 36 are
common multiples.
The LCM is 12.
ANSWER The trains will both return to the station in 12 minutes.
EXAMPLE
2
Finding the Least Common Multiple
Find the least common multiple of 7 and 8.
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, . . .
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, . . .
ANSWER The least common multiple of 7 and 8 is 56.
Lesson 4.4
Least Common Multiple
175
Page 2 of 4
Using Prime Factorization Another way to find the least common
multiple of two or more numbers is to use prime factorization.
EXAMPLE
with
Solving
3
Using Prime Factorization to Find the LCM
Find the LCM of 84 and 360 using prime factorization.
Use the prime
factorization method to
find the least common
multiple of large numbers.
Begin by writing the prime factorization of each number.
84
360
12 7
18 20
4 3 7
2 9 4 5
2 2 3 7
2 3 3 2 2 5
84 2 2 3 7
360 2 2 2 3 3 5
Circle the common factors. Then multiply the common factors (one for
each pair) and all the uncircled factors.
2 2 2 3 3 5 7 23 32 5 7 2520
ANSWER The least common multiple of 84 and 360 is 2520.
Your turn now
Find the LCM of the numbers by listing multiples.
1. 3, 5
2. 12, 16
3. 9, 10
4. 2, 6, 14
Find the LCM of the numbers using prime factorization.
5. 36, 72
EXAMPLE
6. 24, 30
4
7. 54, 126
8. 20, 22, 55
Using the Least Common Multiple
Tour Bus Schedules Three tour buses leave the visitor’s center at
9:00 A.M. Bus A returns to the visitor’s center every 60 minutes, Bus B
returns every 40 minutes, and Bus C returns every 75 minutes. What is
the next time the buses will all return to the visitor’s center?
Solution
Find the least common multiple of 60, 40, and 75.
60 22 3 5
40 23 5
75 3 52
The least common multiple is 23 3 52 600.
ANSWER The buses all return in 600 minutes, or 10 hours,
after 9:00 A.M., which is 7:00 P.M.
176
Chapter 4
Number Patterns and Fractions
Page 3 of 4
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 708
Getting Ready to Practice
1. Vocabulary What is the difference between finding the least common
multiple and finding the greatest common factor of two numbers?
Match the number pairs with their least common multiple.
2. 4, 18
3. 8, 9
4. 6, 21
5. 5, 12
A. LCM 60
B. LCM 72
C. LCM 36
D. LCM 42
Find the LCM of the numbers using prime factorization.
6. 28, 60
7. 49, 56
8. 25, 70
10. Find the Error Describe and correct
9. 22, 64
6 16 96
the error in finding the least common
multiple of 6 and 16.
So, the LCM of
6 and 16 is 96.
11. Running Laps David and James are running laps on a quarter mile
track. It takes James 3 minutes and David 4 minutes to run once
around the track. They both start running from the starting line at the
same time. When will they both be at the starting line again?
Practice and Problem Solving
with
Example
1
2
3
4
Homework
Exercises
12–19
12–19
20–27, 29–36
37
Online Resources
Find the LCM of the numbers by listing multiples.
12. 9, 24
13. 12, 18
14. 16, 20
15. 30, 33
16. 5, 8, 12
17. 6, 11, 18
18. 9, 14, 21
19. 7, 20, 35
Find the LCM of the numbers using prime factorization.
20. 34, 52
21. 28, 46
22. 36, 81
23. 27, 48
24. 42, 56, 140
25. 39, 52, 169
26. 28, 40, 144
27. 16, 25, 27
CLASSZONE.COM
• More Examples
• eTutorial Plus
28. Choose a Strategy You want to find the LCM of 32 and 49. Would you
list multiples or use prime factorization? Explain your choice.
Find the GCF and the LCM of the numbers using prime factorization.
29. 90, 165
30. 34, 66
31. 54, 132
32. 72, 168
33. 288, 405
34. 42, 81, 105
35. 55, 88, 220
36. 78, 96, 174
Lesson 4.4
Least Common Multiple
177
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37. Mayan Calendars The Mayans used more than one calendar system.
History
One calendar had 365 days. Another calendar, considered sacred to the
Mayans, had 260 days. If both calendars began on the same day, in how
many years would they next begin on the same day?
Extended Problem Solving In Exercises 38–40, use the following
information. Cicadas, which are flying insects, live underground until they
are fully developed. It takes 13 years for one type of cicada and 17 years
for another type of cicada to fully develop. In 1998, the two types of cicadas
emerged together.
38. Evaluate In how many years will the two types emerge together again?
39. Analyze How is the answer found in Exercise 38 related to 13 and 17?
40. Predict Between the years 1998 and 2998, how many times will the
two cicadas emerge together? Identify in which years they will emerge.
Algebra Find the LCM of the variable expressions.
■
41. w 2, w 3
Mayan Calendars
Some people believe that
the Temple of Kukulcan,
a Mayan ruin, is a
representation of the solar
calendar. The temple has
4 sides. Each side has
91 steps that lead up to
the top. Counting the top
as 1 step, how many steps
are there altogether?
42. 3d, 9d
43. 4s 2, 2s 4
44. 12x 2, 16x
45. Challenge The greatest common factor of two numbers is 1. Is the
least common multiple of the numbers always the product of the
numbers? Give examples to explain your reasoning.
Mixed Review
46. Find the perimeter and the area of a rectangle with a width of
8 centimeters and a length of 12 centimeters. (Lesson 1.6)
Write two fractions that are equivalent to the given fraction.
(Lesson 4.3)
16
47. 24
12
48. 20
10
49. 25
6
50. 7
Basic Skills Write the number in standard form.
51. one hundred twenty-three
52. four hundred one
53. sixteen hundred forty
54. twenty-two thousand, forty-five
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
178
Chapter 4
55. Extended Response Sarah and Jen are swimming laps. Sarah swims
7 laps in 5 minutes, and Jen swims 11 laps in 6 minutes. If they start
swimming and stop swimming at the same time and they swim a
whole number of laps, what is the least possible amount of time they
could have been swimming? In this amount of time, how many laps
did each girl swim?
Number Patterns and Fractions