8-1 Factors and Greatest Common Factors
Objectives
Write the prime factorization of
numbers.
Find the GCF of monomials.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
Vocabulary
prime factorization
greatest common factor
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
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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Holt McDougal Algebra 1
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8-1 Factors and Greatest Common Factors
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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Holt McDougal Algebra 1
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8-1 Factors and Greatest Common Factors
Remember!
A prime number has exactly two factors, itself
and 1. The number 1 is not prime because it only
has one factor.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
Example 1: Writing Prime Factorizations
Write the prime factorization of 98.
Method 1 Factor tree
Method 2 Ladder diagram
Choose any two factors
Choose a prime factor of 98
of 98 to begin. Keep finding
to begin. Keep dividing by
factors until each branch
prime factors until the
ends in a prime factor.
quotient is 1.
98
2 98
7 49
2 Ÿ 49
7 7
Ÿ
7
7
1
98 = 2 Ÿ 7 Ÿ 7
98 = 2 Ÿ 7 Ÿ 7
The prime factorization of 98 is 2 Ÿ 7 Ÿ 7 or 2 Ÿ 72.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
Check It Out! Example 1
Write the prime factorization of each number.
a. 40
40
2 Ÿ 20
2 Ÿ 10
2 Ÿ 5
40 = 23 Ÿ 5
The prime factorization
of 40 is 2 Ÿ 2 Ÿ 2 Ÿ 5 or
23 Ÿ 5.
Holt McDougal Algebra 1
b. 33
11 33
3
33 = 3 Ÿ 11
The prime factorization
of 33 is 3 Ÿ 11.
8-1 Factors and Greatest Common Factors
Check It Out! Example 1
Write the prime factorization of each number.
c. 49
d. 19
49
7 Ÿ 7
49 = 7 Ÿ 7
The prime factorization
of 49 is 7 Ÿ 7 or 72.
Holt McDougal Algebra 1
1 19
19
19 = 1 Ÿ 19
The prime factorization
of 19 is 1 Ÿ 19.
8-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
Example 2B: Finding the GCF of Numbers
Find the GCF of each pair of numbers.
26 and 52
Method 2 Prime factorization.
26 =
2 Ÿ 13
52 = 2 Ÿ 2 Ÿ 13
2 Ÿ 13 = 26
Write the prime
factorization of each
number.
Align the common
factors.
The GCF of 26 and 52 is 26.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
Check It Out! Example 2a
Find the GCF of each pair of numbers.
12 and 16
Method 1 List the factors.
factors of 12: 1, 2, 3, 4, 6, 12
List all the factors.
factors of 16: 1, 2, 4, 8, 16
Circle the GCF.
The GCF of 12 and 16 is 4.
Holt McDougal Algebra 1
8-1 Factors and Greatest Common Factors
Check It Out! Example 2b
Find the GCF of each pair of numbers.
15 and 25
Method 2 Prime factorization.
15 = 1 Ÿ 3 Ÿ 5
25 = 1 Ÿ 5 Ÿ 5
1 Ÿ
5=5
Write the prime
factorization of each
number.
Align the common
factors.
The GCF of 15 and 25 is 5.
Holt McDougal Algebra 1