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Factors
and
Greatest
and Greatest Common Factors
8-1
8-1 Factors
Common Factors
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
11
8-1 Factors and Greatest Common Factors
Warm Up
Tell whether the second number is a factor
of the first number
1. 50, 6
no
2. 105, 7
3. List the factors of 28.
±14, ±28
yes
±1, ±2, ±4, ±7,
Tell whether each number is prime or
composite. If the number is composite, write
it as the product of two numbers.
4. 11 prime
Holt Algebra 1
5. 98 composite; 49  2
8-1 Factors and Greatest Common Factors
Objectives
Write the prime factorization of
numbers.
Find the GCF of monomials.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Vocabulary
prime factorization
greatest common factor
Holt 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 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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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 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  49
7 7

7
7
1
98 = 2  7  7
98 = 2  7  7
The prime factorization of 98 is 2  7  7 or 2  72.
Holt 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  20
2  10
2  5
40 = 23  5
The prime factorization
of 40 is 2  2  2  5 or
23  5.
Holt Algebra 1
b. 33
11 33
3
33 = 3  11
The prime factorization
of 33 is 3  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  7
49 = 7  7
The prime factorization
of 49 is 7  7 or 72.
Holt Algebra 1
1 19
19
19 = 1  19
The prime factorization
of 19 is 1  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 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 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  13
52 = 2  2  13
2  13 = 26
Write the prime
factorization of each
number.
Align the common
factors.
The GCF of 26 and 52 is 26.
Holt 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 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  3  5
25 = 1  5  5
1
Holt Algebra 1
5=5
Write the prime
factorization of each
number.
Align the common
factors.
8-1 Factors and Greatest Common Factors
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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Example 3A: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
15x3 and 9x2
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
3
Write the prime factorization of
each coefficient and write
powers as products.
Align the common factors.
x  x = 3x2 Find the product of the common
factors.
The GCF of 3x3 and 6x2 is 3x2.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Example 3B: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
8x2 and 7y3
Write the prime
factorization of each
8x2 = 2  2  2 
xx
coefficient and write
7y3 =
7
y  y  y powers as products.
Align the common
factors.
The GCF 8x2 and 7y is 1.
Holt Algebra 1
There are no
common factors
other than 1.
8-1 Factors and Greatest Common Factors
Helpful Hint
If two terms contain the same variable raised to
different powers, the GCF will contain that
variable raised to the lower power.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Check It Out! Example 3a
Find the GCF of each pair of monomials.
18g2 and 27g3
18g2 = 2  3  3 
27g3 =
gg
Write the prime factorization
of each coefficient and
write powers as products.
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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
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 =
The GCF of 16a6 and 7b is 1.
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33b
Align the common
factors.
There are no common factors
other than 1.
8-1 Factors and Greatest Common Factors
Check It Out! Example 3c
Find the GCF of each pair of monomials.
8x and 7v2
8x = 2  2  2  x
7v2 =
7vv
The GCF of 8x and 7v2 is 1.
Holt Algebra 1
Write the prime factorization
of each coefficient and
write powers as products.
Align the common factors.
There are no common
factors other than 1.
8-1 Factors and Greatest Common Factors
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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Example 4 Continued
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Find the common
factors of 18
and 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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Example 4 Continued
18 chocolate milk cartons
= 3 rows
6 containers per row
24 regular milk cartons
6 containers per row
= 4 rows
When the greatest possible number of types of
milk is in each row, there are 7 rows in total.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Check It Out! Example 4 Continued
Find the common
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 factors of 36
and 48.
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.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
36 pop CDs
12 CDs per shelf
= 3 shelves
48 country CDs
12 CDs per shelf
= 4 shelves
When the greatest possible number of CD types
are on each shelf, there are 7 shelves in total.
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Lesson Quiz: Part 1
Write the prime factorization of each number.
1. 50
2  52
2. 84
22  3  7
Find the GCF of each pair of numbers.
3. 18 and 75 3
4. 20 and 36 4
Holt Algebra 1
8-1 Factors and Greatest Common Factors
Lesson Quiz: Part II
Find the GCF each pair of monomials.
5. 12x and 28x3 4x
6. 27x2 and 45x3y2 9x2
7. Cindi is planting a rectangular flower bed with 40
orange flower and 28 yellow flowers. She wants
to plant them so that each row will have the
same number of plants but of only one color. How
many rows will Cindi need if she puts the greatest
possible number of plants in each row?
17
Holt Algebra 1