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4-8 Least Common Multiple - Glencoe

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4-8
Least Common Multiple
MAIN IDEA
Find the least common
multiple of two or more
numbers.
New Vocabulary
multiple
least common multiple
(LCM)
Use cubes to build the first row of each prism as shown.
1. Add a second row to each prism. Record
the total number of cubes used in a table
like the one shown below.
Number of Rows
Cubes in Prism A
Cubes in Prism B
1
4
6
2
3
8
12
1SJTN"
4
12
18
16
24
Math Online
glencoe.com
2. Add rows until each prism has four rows.
• Extra Examples
• Personal Tutor
• Self-Check Quiz
3. Describe two prisms that have the same
1SJTN#
number of cubes. See margin.
4. If you keep adding rows, will the two prisms have the same
number of cubes again? See margin.
A multiple is the product of a number and any whole number. The
least common multiple, or LCM, of two or more numbers is the least of
their common multiples, excluding zero.
Find the LCM
1 Find the LCM of 6 and 10.
List the nonzero multiples.
METHOD 1
List the multiples of 6 until you come to a number that is also a
multiple of 10.
multiples of 6: 6, 12, 18, 24, 30, …
multiples of 10: 10, 20, 30, …
Notice that 30 is also a multiple of 10. The LCM of 6 and 10 is 30.
Vocabulary Link
Multiply
Everyday Use to find the
product
Multiple
Math Use the product of
a number and any whole
number
METHOD 2
6=2В·3
10 = 2 В· 5
Use prime factorization.
The prime factors of 6
and 10 are 2, 3, and 5.
The LCM is the least product that contains the prime factors of each
number. So, the LCM of 6 and 10 is 2 В· 3 В· 5 or 30.
Lesson 4-8 Least Common Multiple
211_0211_0214_CH04_L8_874046 211
211
9/21/07 9:35:00 AM
2 Find the LCM of 45 and 75.
Use Method 2. Find the prime factorization of each number.
45 = 3 В· 3 В· 5 or 3 2 В· 5
75 = 3 В· 5 В· 5 or 3 В· 5
The prime factors of 45 and 75
are 3 and 5. Write the prime
factorization using exponents.
2
The LCM is the product of the prime factors 3 and 5, with each one
raised to the highest power it occurs in either prime factorization.
The LCM of 45 and 75 is 32 В· 5 2, which is 225.
Find the LCM of each set of numbers.
a. 3, 12
b. 10, 12
c. 25, 30
3 PARTY Ling needs to buy paper plates, napkins, and cups for a
party. Plates come in packages of 12, napkins come in packages of
16, and cups come in packages of 8. What is the least number of
packages she will have to buy if she wants to have the same
number of plates, napkins, and cups?
First find the LCM of 8, 12, and 16.
8 = 2 В· 2 В· 2 or 2 3
12 = 2 В· 2 В· 3 or 2 2 В· 3
16 = 2 В· 2 В· 2 В· 2 or 2
4
The prime factors of 8, 12, and
16 are 2 and 3. Write the prime
factorization using exponents.
The LCM of 8, 12, and 16 is 2 4 В· 3, which is 48.
To find the number of packages of each Ling needs to buy, divide
48 by the amount in each package.
cups: 48 Г· 8 or 6 packages
plates: 48 Г· 12 or 4 packages
napkins: 48 Г· 16 or 3 packages
So, Ling will need to buy 6 packages of cups, 4 packages of plates,
and 3 packages of napkins.
d. VEHICLES Mr. Hernandez changes his car’s oil every 3 months,
rotates the tires every 6 months, and replaces the air filter once a
year. If he completed all three tasks in April, what will be the next
month he again completes all three tasks?
212
Chapter 4 Fractions, Decimals, and Percents
211_214_C4_L8_892328.indd 212
4/8/10 10:01:06 AM
Examples 1–3
(pp. 211–212)
Find the LCM of each set of numbers.
28
1. 4, 14
4. 21, 35
Example 3
(p. 212)
HOMEWORK
For
Exercises
8–13, 20
14–19, 21
HELP
See
Examples
1, 2
3
105
42
3. 12, 15
60
5. 3, 5, 12
60
42
6. 6, 14, 21
7. GOVERNMENT The number of years per term for a
U.S. President, senator, and representative is shown.
Suppose a senator was elected in the presidential
election year 2008. In what year will he or she
campaign again during a presidential election year?
2020
Elected Office Term (yr)
President
4
Senator
6
Representative
2
Find the LCM for each set of numbers.
8. 6, 8
24
9. 8, 18
72
10. 12, 16
48
315
11. 24, 36
72
12. 11, 12
132
13. 45, 63
14. 2, 3, 5
30
15. 6, 8, 9
72
16. 8, 12, 16
17. 12, 15, 28
Exercise Levels
A: 8–21
B: 22–30
C: 31–34
2. 6, 7
420
18. 22, 33, 44
132
48
19. 12, 16, 36
144
20. CHORES Hernando walks his dog every
two days. He gives his dog a bath once a
week. Today, Hernando walked his dog
and then gave her a bath. How many
days will pass before he does both
chores on the same day? 14 days
Friend
21. TEXT MESSAGING Three friends use text
messaging to notify their parents of their
whereabouts. If all three contact their parents
at 3:00 p.m., at what time will all three
contact their parents again at the same time?
6:00 P.M.
Find the LCM of each set.
22. $3.00, $14.00
$42
23. 10Вў, 25Вў, 5Вў
50Вў
Time Interval
Linda
every 30 min
Brandon
every 45 min
Edward
every 60 min
24. 9 inches, 2 feet
72 in. or 6 ft
Write two numbers whose LCM is the given number.
25–28. Sample
answers are given.
25. 35
5, 7
26. 56
7, 8
27. 70
10, 35
28. 30
6, 15
29. SNACKS Alvin’s mom needs to buy snacks for soccer practice. Juice boxes
EXTRA
PRACTICE
See pages 678, 707.
come in packages of 10. Oatmeal snack bars come in packages of 8.
She wants to have the same number of juice boxes and snack bars, what is
the least number of packages of each snack that she will have to buy?
4 packages of juice boxes and 5 packages of oatmeal snack bars
30. REASONING The LCM of two consecutive positive numbers is greater than
200 and is a multiple of 7. What are the least possible numbers? 14 and 15
Lesson 4-8 Least Common Multiple
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H.O.T. Problems
31. CHALLENGE Two numbers have a GCF of 3 В· 5. Their LCM is 22 В· 3 В· 5. If one
32. Number
sense; Sample
answer: The LCM
of 14, 16, and 8 is
112. So, 112
minutes later, or
at 1:27 P.M.
32. SELECT A TECHNIQUE The schedule for
of the numbers is 3 В· 5, what is the other number? 2 2 В· 3 В· 5, or 60
Clark Street Train Station
each of three trains is shown. Suppose a
Train
train from each line leaves Clark Street
Red-line
at 11:35 a.m. Which of the following
Blue-line
technique(s) might you use to determine
Brown-line
the next time all three trains will be leaving
at the same time? Justify your selection(s).
Then use the technique to solve the problem.
mental math
number sense
Leaves Station
every 14 minutes
every 16 minutes
every 8 minutes
estimation
33. OPEN ENDED Write three numbers that have an LCM of 30.
Sample answer: 3, 10, 15
34.
WR ITING IN MATH Describe the relationship between 4, 20, and 5 using
the words factor and multiple. 4 and 5 are factors of 20; 20 is a multiple (and
LCM) of 4 and 5.
36. SHORT RESPONSE Wil swims every
35. Which rule describes the common
multiples of 12 and 18, where n
represents the counting numbers? C
third day, runs every fourth day,
and lifts weights every fifth day.
If Wil does all three activities today,
how many days will pass before he
does all three activities on the same
day again? 60
A 12n
B 18n
C 36n
D 216n
Write each percent as a decimal.
37. 55%
0.55
38. 26.4%
(Lesson 4-7)
0.264
39.
_1 % 0.0025
4
0.02
40. 2%
41. DIAMONDS Sixty-eight percent of engagement rings have a diamond that is
round in shape. Write this percent as a fraction in simplest form.
42. ALGEBRA Solve 3x = 18. (Lesson 3-3)
(Lesson 4-6)
17
_
25
6
43. ALGEBRA Rose swam 7 laps more than twice the number of laps her sister
swam. Write an algebraic expression to represent this situation.
(Lesson 3-1)
PREREQUISITE SKILL Replace each в—Џ with <, > or = to make a true sentence.
44. 6.85 в—Џ 5.68
214
>
45. 2.34 в—Џ 2.43
<
46. 6.9 в—Џ 5.99
2s + 7
(Page 670)
>
Chapter 4 Fractions, Decimals, and Percents
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9/21/07 9:35:14 AM
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