The whole numbers that are multiplied to find a product are called factors of that product' A number

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The whole numbers that are multiplied to find a
product are called factors of that product. A
number is divisible by its factors.
You can use the factors of a number to write the
number as a product. The number 12 can be
factored several ways.
Factorizations of 12
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The order of factors does not change the product,
but there is only one example below that cannot
be factored further. The circled factorization is
the prime factorization because all the factors
are prime numbers. The prime factors can be
written in any order, and except for changes in
the order, there is only one way to write the
prime factorization of a number.
Factorizations of 12
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Example 1 Writing Prime Factorizations
Write the prime factorization of 98.
Method 1 Factor tree
Method 2 Ladder diagram
Choose any two factors of 98 to begin. Keep
finding factors until each branch ends in a prime
factor.
Choose a prime factor of 98 to begin. Keep
dividing by prime factors until the quotient is 1.
The prime factorization of 98 is 2 ? 7 ? 7 or 2
? 72.
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Check It Out! Example 1
Write the prime factorization of each number.
a. 40
b. 33
40 23 ? 5
33 3 ? 11
The prime factorization of 40 is 2 ? 2 ? 2 ? 5
or 23 ? 5.
The prime factorization of 33 is 3 ? 11.
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Check It Out! Example 1
Write the prime factorization of each number.
c. 49
d. 19
49 7 ? 7
19 1 ? 19
The prime factorization of 49 is 7 ? 7 or 72.
The prime factorization of 19 is 1 ? 19.
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Factors that are shared by two or more whole
numbers are called common factors. The greatest
of these common factors is called the greatest
common factor, or GCF.
Factors of 12 1, 2, 3, 4, 6, 12
Factors of 32 1, 2, 4, 8, 16, 32
Common factors 1, 2, 4
The greatest of the common factors is 4.
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Example 2A Finding the GCF of Numbers
Find the GCF of each pair of numbers.
100 and 60
Method 1 List the factors.
factors of 100 1, 2, 4, 5, 10, 20, 25, 50, 100
List all the factors.
factors of 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20,
30, 60
Circle the GCF.
The GCF of 100 and 60 is 20.
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Example 2B Finding the GCF of Numbers
Find the GCF of each pair of numbers.
26 and 52
Method 2 Prime factorization.
Write the prime factorization of each number.
26 2 ? 13
52 2 ? 2 ? 13
Align the common factors.
2 ? 13 26
The GCF of 26 and 52 is 26.
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Check It Out! Example 2a
Find the GCF of each pair of numbers.
12 and 16
Method 1 List the factors.
List all the factors.
factors of 12 1, 2, 3, 4, 6, 12
Circle the GCF.
factors of 16 1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.
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Check It Out! Example 2b
Find the GCF of each pair of numbers.
15 and 25
Method 2 Prime factorization.
Write the prime factorization of each number.
15 1 ? 3 ? 5
25 1 ? 5 ? 5
Align the common factors.
1 ? 5 5
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You can also find the GCF of monomials that
include variables. To find the GCF of monomials,
write the prime factorization of each coefficient
and write all powers of variables as products.
Then find the product of the common factors.
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Example 3A Finding the GCF of Monomials
Find the GCF of each pair of monomials.
15x3 and 9x2
Write the prime factorization of each coefficient
and write powers as products.
15x3 3 ? 5 ? x ? x ? x
9x2 3 ? 3 ? x ? x
Align the common factors.
3 ? x ? x 3x2
Find the product of the common factors.
The GCF of 15x3 and 9x2 is 3x2.
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Example 3B Finding the GCF of Monomials
Find the GCF of each pair of monomials.
8x2 and 7y3
Write the prime factorization of each coefficient
and write powers as products.
8x2 2 ? 2 ? 2 ? x ? x
7y3 7 ? y ? y ? y
Align the common factors.
There are no common factors other than 1.
The GCF 8x2 and 7y3 is 1.
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Check It Out! Example 3a
Find the GCF of each pair of monomials.
18g2 and 27g3
Write the prime factorization of each coefficient
and write powers as products.
18g2 2 ? 3 ? 3 ? g ? g
27g3 3 ? 3 ? 3 ? g ? g ? g
Align the common factors.
3 ? 3 ? g ? g
Find the product of the common factors.
The GCF of 18g2 and 27g3 is 9g2.
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Check It Out! Example 3b
Find the GCF of each pair of monomials.
Write the prime factorization of each coefficient
and write powers as products.
16a6 and 9b
16a6 2 ? 2 ? 2 ? 2 ? a ? a ? a ? a ? a ? a
9b
3 ? 3 ? b
Align the common factors.
The GCF of 16a6 and 9b is 1.
There are no common factors other than 1.
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Check It Out! Example 3c
Find the GCF of each pair of monomials.
8x and 7v2
Write the prime factorization of each coefficient
and write powers as products.
8x 2 ? 2 ? 2 ? x
7v2 7 ? v ? v
Align the common factors.
There are no common factors other than 1.
The GCF of 8x and 7v2 is 1.
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Example 4 Application
A cafeteria has 18 chocolate-milk cartons and 24
regular-milk cartons. The cook wants to arrange
the cartons with the same number of cartons in
each row. Chocolate and regular milk will not be
in the same row. How many rows will there be if
the cook puts the greatest possible number of
cartons in each row?
The 18 chocolate and 24 regular milk cartons must
be divided into groups of equal size. The number
of cartons in each row must be a common factor of
18 and 24.
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Example 4 Continued
Find the common factors of 18 and 24.
Factors of 18 1, 2, 3, 6, 9, 18
Factors of 24 1, 2, 3, 4, 6, 8, 12, 24
The GCF of 18 and 24 is 6.
The greatest possible number of milk cartons in
each row is 6. Find the number of rows of each
type of milk when the cook puts the greatest
number of cartons in each row.
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Example 4 Continued
When the greatest possible number of types of
milk is in each row, there are 7 rows in total.
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Check It Out! Example 4
Adrianne is shopping for a CD storage unit. She
has 36 CDs by pop music artists and 48 CDs by
country music artists. She wants to put the same
number of CDs on each shelf without putting pop
music and country music CDs on the same shelf. If
Adrianne puts the greatest possible number of CDs
on each shelf, how many shelves does her storage
unit need?
The 36 pop and 48 country CDs must be divided
into groups of equal size. The number of CDs in
each row must be a common factor of 36 and 48.
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Check It Out! Example 4 Continued
Find the common factors of 36 and 48.
Factors of 36 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The GCF of 36 and 48 is 12.
The greatest possible number of CDs on each shelf
is 12. Find the number of shelves of each type of
CDs when Adrianne puts the greatest number of CDs
on each shelf.
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When the greatest possible number of CD types are
on each shelf, there are 7 shelves in total.