Transcript Find the least common multiple

2-8 Least Common Multiple
JANUARY 20, 2012
Warm-Up
LCM cont.
Word Problems
Exit Ticket
2-8 Least Common Multiple
Friday, January 20: Warm-Up
List the two strategies you can
use to find the Greatest
Common Factor (GCF).
2-8 Least Common Multiple
4-5 Least Common Multiple
Goal: Learn how to find the least common
multiple of two or more numbers.
2-8 Least Common Multiple
Vocabulary
multiple
common multiple
least common multiple
2-8 Least Common Multiple
A multiple of a number is a product of that number
and a whole number. Some multiples of 7,500 and
5,000 are as follows:
7,500: 7,500, 15,000, 22,500, 30,000, 37,500, 45,000, . . .
5,000: 5,000, 10,000, 15,000, 20,000, 25,000, 30,000, . . .
A common multiple of two or more numbers
is a number that is a multiple of each of the
given numbers. So 15,000 and 30,000 are
common multiples of 7,500 and 5,000.
2-8 Least Common Multiple
least common
multiple
(LCM)
twonumber
or
A The
multiple
of a number
is a product
of of
that
more
numbers
is theSome
common
multiple
with the
and
a whole
number.
multiples
of 7,500
and
least are
value.
5,000
as follows:
7,500: 7,500, 15,000, 22,500, 30,000, 37,500, 45,000, . . .
Example: The LCM of 7,500 and 5,000 is 15,000.
5,000: 5,000, 10,000, 15,000, 20,000, 25,000, 30,000, . . .
A common multiple of two or more numbers
is a number that is a multiple of each of the
given numbers. So 15,000 and 30,000 are
common multiples of 7,500 and 5,000.
2-8 Least Common Multiple
Example 1: Using a List to Find the LCM
Find the least common multiple (LCM).
A. 2, 7
Multiples of 2: 2, 4, 6, 8, 10, 12, 14
Multiples of 7: 7, 14, 21, 28, 35
The LCM is 14.
List some multiples of
each number.
Find the least value that
is in both lists.
B. 3, 6, 9
Multiples of 3: 3, 6, 9, 12, 15, 18, 21
Multiples of 6: 6, 12, 18, 24, 30
Multiples of 9: 9, 18, 27, 36, 45
The LCM is 18.
List some multiples of
each number.
Find the least value that
is in all the lists.
2-8 Least Common Multiple
On Your Own: Example 2
Find the least common multiple (LCM).
A. 3, 7
Multiples of 3: 3, 6, 9, 12, 15, 18, 21
Multiples of 7: 7, 14, 21, 28
The LCM is 21.
B. 2, 6, 4
Multiples of 2: 2, 4, 6, 8, 10, 12, 14
Multiples of 6: 6, 12, 18
Multiples of 4: 4, 8, 12
The LCM is 12.
List some multiples of
each number.
Find the least value
that is in both lists.
List some multiples of
each number.
Find the least value
that is in all the lists.
2-8 Least Common Multiple
Example 3: Using Prime Factorization to Find the
LCM
Find the least common multiple (LCM).
60, 130
60 = 2 · 2 · 3 · 5
Write the prime factorization of
each number.
130 = 2 · 5 · 13
Circle the common prime factors.
2, 2, 3, 5, 13
List the prime factors, using
the circled factors only once.
2 · 2 · 3 · 5 · 13
Multiply the factors in the list.
The LCM is 780.
2-8 Least Common Multiple
Example 4: Using Prime Factorization to Find the
LCM
Find the least common multiple (LCM).
14, 35, 49
14 = 2 · 7
35 = 5 · 7
49 = 7 · 7
Write the prime factorization of
each number.
Circle the common prime factors.
2, 5, 7, 7
List the prime factors, using
the circled factors only once.
2·5·7·7
Multiply the factors in the list.
The LCM is 490.
2-8 Least Common Multiple
On Your Own: Example 5
Find the least common multiple (LCM).
50, 110
50 = 2 · 5 · 5
Write the prime factorization of
each number.
110 = 2 · 5 · 11
Circle the common prime factors.
2, 5, 5, 11
List the prime factors, using
the circled factors only once.
2 · 5 · 5 · 11
Multiply the factors in the list.
The LCM is 550.
2-8 Least Common Multiple
On Your Own: Example 6
18, 36, 54
18 = 2 · 3 · 3
Write the prime factorization of
each number.
36 = 2 · 2 · 3 · 3
Circle the common prime factors.
54 = 2 · 3 · 3 · 3
2, 2, 3, 3, 3
List the prime factors, using
the circled factors only once.
2·2·3·3·3
Multiply the factors in the list.
The LCM is 108.
2-8 Least Common Multiple
Word Problems With GCF and LCM
Steps:
1.Understand the Problem
2.Make a Plan
3.Solve
4.Look Back
2-8 Least Common Multiple
Example 1: Word Problem
You have 120 red beads, 100 white
beads, and 45 blue beads. You want to
use all the beads to make bracelets that
have red, white, and blue beads on each.
What is the greatest number of
matching bracelets you can make?
2-8 Least Common Multiple
Additional Example 3 Continued
1
Understand the Problem
Rewrite the question as a statement.
• Find the greatest number of matching bracelets
you can make.
List the important information:
• There are 120 red beads, 100 white beads,
and 45 blue beads.
• Each bracelet must have the same
number of red, white, and blue beads.
The answer will be the GCF of 120, 100, and 45.
2-8 Least Common Multiple
2
Additional Example 3 Continued
Make a Plan
You can list the prime factors of 120, 100,
and 45 to find the GCF.
3
Solve
120 = 2 · 2 · 2 · 3 · 5
100 = 2 · 2 · 5 · 5
45 = 3 · 3 · 5
The GCF of 120, 100, and 45 is 5.
You can make 5 bracelets.
2-8 Least Common Multiple
Additional Example 3 Continued
4
Look Back
If you make 5 bracelets, each one will have
24 red beads, 20 white beads, and 9 blue
beads, with nothing left over.
2-8 Least Common Multiple
Example 2: Word Problem
Mr. Washington will set up the band chairs all
in rows of 6 or all in rows of 8. What is the
least number of chairs he will set up?
Find the LCM of 6 and 8.
6=2·3
8=2·2·2
The LCM is 2 · 2 · 2 · 3 = 24.
He will set up at least 24 chairs.
2-8 Least Common Multiple
On Your Own: Example 3
Two satellites are put into orbit over the same
location at the same time. One orbits the earth
every 24 hours, while the second completes an
orbit every 18 hours. How much time will
elapse before they are once again over the
same location at the same time?
Find the LCM of 24 and 18.
24 = 2 · 2 · 2 · 3
18 = 2 · 3 · 3
The LCM is 2 · 2 · 2 · 3 · 3 = 72.
72 hours will elapse before they are over the
same location at the same time.
2-8 Least Common Multiple
Exit Ticket
Find the GCF and tell me what strategy
you used:
1. 28, 40
Find the LCM and tell me what strategy
you used:
2. 4, 6, 15
2-8 Least Common Multiple
2-8 Least Common Multiple
2-8 Least Common Multiple
2-8 Least Common Multiple
Lesson Quiz
Find the least common multiple (LCM).
1. 18, 21
126
2. 24, 27
3. 4, 6, 15
60
4. 4, 8, 16 16
216
5. You are planning a picnic. You can purchase
paper plates in packages of 30, paper napkins
in packages of 50, and paper cups in packages
of 20. What is the least number of each type
of package that you can buy and have an
equal number of each?
10 packages of plates
6 packages of napkins
15 packages of cups
2-8 Least Common Multiple
Lesson Quiz for Student Response Systems
1. Identify the least common multiple (LCM)
of 15 and 24.
A. 105
B. 112
C. 120
D. 124
2-8 Least Common Multiple
Lesson Quiz for Student Response Systems
2. Identify the least common multiple (LCM)
of 21 and 24.
A. 162
B. 168
C. 172
D. 178
2-8 Least Common Multiple
Lesson Quiz for Student Response Systems
3. Identify the least common multiple (LCM)
of 6, 8, and 18.
A. 72
B. 64
C. 56
D. 48
2-8 Least Common Multiple
Lesson Quiz for Student Response Systems
4. Identify the least common multiple (LCM)
of 5, 10, and 20.
A. 25
B. 20
C. 15
D. 10
2-8 Least Common Multiple
Lesson Quiz for Student Response Systems
5. You are planning to distribute pencils,
notebooks, and school bags to children in
an orphanage. You can purchase pencils in
packages of 30, notebooks in packages of
40, and school bags in packages of 50.
What is the least number of each type of
package that you can buy and have an
equal number of each?
A. 20 packages of pencils, 15 packages of
notebooks, and 12 packages of school bags
B. 12 packages of pencils, 15 packages of
notebooks, and 20 packages of school bags