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Lesson 11
Chapter 2
Find Common Factors and Common Multiples
6.NS.B.4
Words to Know
greatest common
factor
least common
multiple
Real-World Connection
Sue and her family are waiting at the airport
for a shuttle to take them to their car. They
need two shuttles, but both shuttles are
leaving as they walk to the shuttle area. One
of the shuttles returns to the airport every
9 minutes. The other shuttle returns every
12 minutes. How long will it be before the
two shuttles are at the airport at the same time again?
Breaking Down the Standard
The greatest common factor (GCF) of a set of numbers is the greatest (largest) factor
common to all the numbers. The numbers could have several factors in common, or
they could have just one. So, to find the GCF, list all of the factors of the numbers.
example
Find the greatest common factor of 12 and 18.
List the factors of each number.
• Factors of 12:
Write multiplication expressions that have a product of 12:
1 ⫻ 12
List the factors:
1
2⫻6
2
3
4
6
3⫻4
12
• Factors of 18:
Write multiplication expressions that have a product of 18:
1 ⫻ 18
2⫻9
3⫻6
1
2
3
6
9
18
Look for factors in each list that are the same, or “common.”
1, 2, 3, 4, 6, 12
1, 2, 3, 6, 9, 18
The greatest common factor is 6.
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List the factors:
Find Common Factors and Common Multiples
Lesson 11
The distributive property states that for any numbers represented by a, b, and c,
a(b ⫹ c) ⫽ ab ⫹ ac. For example, 17(15 ⫹ 29) ⫽ 17 ⫻ 15 ⫹ 17 ⫻ 29. This property and
GCF often are used together.
example
Rewrite 16 ⫹ 24 using the distributive property and GCF.
Begin by finding the greatest common factor of 16 and 24.
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The greatest common factor is 8.
Use the distributive property to express 16 ⫹ 24 as 8(2 ⫹ 3).
Remember, 8(2 ⫹ 3) ⫽ 8 ⫻ 2 ⫹ 8 ⫻ 3 ⫽ 16 ⫹ 24.
Also, 8(2 ⫹ 3) ⫽ 8 ⫻ 5 ⫽ 40, so 16 ⫹ 24 ⫽ 40.
The least common multiple (LCM) of a set of numbers is the least (smallest) positive
multiple common to all the numbers. Every number has an infinite number of multiples.
example
Find the least common multiple of 4 and 6.
Hint, Hint . . .
Listing multiples of a
number is like skipcounting by that
number.
Make a list of the first few multiples of each number.
• Multiples of 4:
4
8
• Multiples of 6:
6
12
12
You can stop when you find a multiple in common. There are other
common multiples, such as 24, 36, and 48, but the smallest, or “least”
common multiple is 12.
Now solve the Real-World Connection problem.
Two shuttles leave the airport at the same time. One of the shuttles returns to the
airport every 9 minutes. The other shuttle returns every 12 minutes. How long will it be
before the two shuttles are at the airport at the same time again?
To solve this problem, find the least common multiple (LCM) of 9 and 12.
Multiples of 9: 9, 18, 27, 36
Multiples of 12: 12, 24, 36
Copying is prohibited.
The least common multiple is 36. This means that after 36 minutes pass, the shuttles
will be at the airport at the same time again.
The first shuttle will be returning for the fourth time, and the second shuttle will be
returning for the third time.
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Lesson 11
Find Common Factors and Common Multiples
Guided Practice
Hint, Hint . . .
1.
1
When finding
possible factors, list
them in numerical
order. For 48, start at
the outside of the list,
writing 1 on the left
and 48 on the right,
and work your way in.
Write the factors of 72
directly underneath.
Find the greatest common factor of 48 and 72.
List the factors of 48.
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
List the factors of 72.
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Look for the largest factor
that occurs in both lists.
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The greatest common factor of 48 and 72
is 24.
2.
48 ⫽ 4 ⫻ 12
72 ⫽ 6 ⫻ 12
You may have noticed that 48 and 72 are
multiples of 12.
48 ⫽ 2 ⫻ 2 ⫻ 12
72 ⫽ 3 ⫻ 2 ⫻ 12
4 and 6 share a common factor of 2.
48 ⫽ 2 ⫻ (2 ⫻ 12) ⫽ 2 ⫻ 24
72 ⫽ 3 ⫻ (2 ⫻ 12) ⫽ 3 ⫻ 24
24 is the GCF.
Find the least common multiple of 4 and 9.
List several multiples of 4.
4, 8, 12, 16, 20, 24, 28, 32, 36
List several multiples of 9.
9, 18, 27, 36
Look for the smallest number
that occurs in both lists.
36 is the LCM.
Check that your answer
makes sense.
4 and 9 have no common factors other than
1, so the first common multiple will be 4 ⫻ 9.
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4 ⫻ 9 ⫽ 36
Find Common Factors and Common Multiples
Lesson 11
Guided Application
1.
Select the three correct statements about the factors and/or multiples of the
following pairs of numbers.
The numbers 8 and 12 have three common factors.
B
The greatest common factor of 12 and 28 is 2.
C
The least common multiple of 5 and 25 is 25.
D
The greatest common factor of 35 and 100 is 5.
E
Using the distributive property, 12 ⫹ 15 can be rewritten as 3(3 ⫹ 5).
A
The numbers 8 and 12 have
three common factors.
Correct. 8 and 12 have common factors
of 1, 2, and 4.
B
The greatest common factor
of 12 and 28 is 2.
Incorrect. While 2 is a common factor
of 12 and 28, the greatest common
factor is 4:
4 ⫻ 3 ⫽ 12, 4 ⫻ 7 ⫽ 28.
C
The least common multiple
of 5 and 25 is 25.
Correct. 25 and 5 are multiples of 5:
25 ⫽ 5 ⫻ 5, 5 ⫽ 5 ⫻ 1. Sometimes, one
number will be a multiple of the other.
D
The greatest common factor
of 35 and 100 is 5.
Correct. 35 ⫽ 5 ⫻ 7, and 100 ⫽ 5 ⫻ 20.
Since 7 and 20 have no common factors
other than 1, 5 is the GCF.
E
Using the distributive property,
12 ⴙ 15 can be rewritten as
3(3 ⴙ 5).
Incorrect. While the greatest common
factor is 3, the sum inside the
parentheses should be (4 ⫹ 5). The
expression should be 3(4 ⫹ 5) because
3 ⫻ 4 ⫽ 12.
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A
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Lesson 11
Find Common Factors and Common Multiples
Independent Practice
Answer the questions.
1.
Hint, Hint . . .
2.
Which equation is true and uses the GCF of 12 and 30?
Ⓐ
12 ⫹ 30 ⫽ 2(6 ⫹ 15)
Ⓑ
12 ⫹ 30 ⫽ 3(4 ⫹ 10)
Ⓒ
12 ⫹ 30 ⫽ 6(2 ⫹ 5)
Ⓓ
12 ⫹ 30 ⫽ 6(2 ⫹ 7)
Use the numbers in the box to write the greatest common factor of each of
the following pairs of numbers. The numbers can only be used once.
Numbers have
a limited (finite)
number of factors
but an infinite
number of multiples.
1
2
3
5
7
9
27 and 36
11 and 14
21 and 35
3.
Part A
Is 40 the least common multiple of 4 and 10?
Write Yes or No in the box.
4.
Part B
Explain your answer.
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Find Common Factors and Common Multiples
5.
Write a number in the box to make the equation true.
45 ⫹ 30 ⫽
6.
(3 ⫹ 2)
Circle the number that correctly completes the sentence.
The greatest common factor of 24 and 27 is
3
Hint, Hint . . .
.
Factors will always
be less than or equal
to the number.
Multiples will always
be greater than or
equal to the number.
4
6
9
7.
8.
Lesson 11
Circle the two correct statements.
A
The least common multiple of 6 and 7 is 21.
B
The least common multiple of 9 and 5 is 45.
C
The least common multiple of 3 and 6 is 9.
D
The least common multiple of 11 and 12 is 1.
E
The least common multiple of 10 and 8 is 40.
Is each equation written as a multiple of a sum of two numbers correctly?
Select Yes or No.
9.
a.
42 ⫹ 33 ⫽ 3(14 ⫹ 11)
Yes
No
b.
12 ⫹ 9 ⫽ 3(4 ⫹ 3)
Yes
No
c.
26 ⫹ 6 ⫽ 6(4 ⫹ 1)
Yes
No
d.
22 ⫹ 42 ⫽ 22(1 ⫹ 2)
Yes
No
Cindy is having a birthday party, and she wants to make gift bags for her
guests. She has 36 game cards and 60 colored pencils to divide among the
gift bags. She wants to give the same number of each item to each guest.
Write your answers in the boxes.
Copying is prohibited.
What is the greatest number of gift bags she can make?
How many game cards will each guest get?
How many colored pencils will each guest get?
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Lesson 11
Find Common Factors and Common Multiples
10. Part A
What is the greatest common factor of 20 and 16?
Write your answer in the box.
11. Part B
Explain your answer.
Copying is prohibited.
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