lesson 7-1 factors and greatest common factors...holt mcdougal algebra 1 7-1 factors and greatest...

Holt McDougal Algebra 1

7-1 Factors and Greatest Common Factors7-1Factors and Greatest Common Factors

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz



7-1 Factors and Greatest Common Factors

Warm Up

1. 50, 6 2. 105, 7

3. List the factors of 28.

Tell whether each number is prime or composite. If the number is composite, write it as the product of two numbers.

no yes

±1, ±2, ±4, ±7,

Tell whether the second number is a factor of the first number

±14, ±28

4. 11 5. 98 composite; 49 • 2prime



Write the prime factorization of numbers.

Find the GCF of monomials.

Objectives



prime factorization

greatest common factor

Vocabulary



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

••••••••

•••• ••••••••••••••••



Factorizations of 12

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.

••••••••

•••• ••••••••••••••••



A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor.

Remember!



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.

98 = 2 7 7 ��

98

2 49

7 7

�

�

98

49

7

1

2

7

7

98 = 2 7 7 ��

The prime factorization of 98 is 2 � 7 � 7 or 2 � 72.



Check It Out! Example 1

Write the prime factorization of each number.

a. 40

40

2 20

2 10

�

�

2 5�

33

3

11

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.





c. 49 d. 19

49

7 7�19

19

1

49 = 7 � 7

The prime factorization of 49 is 7 � 7 or 72.

19 = 1 � 19

The prime factorization of 19 is 1 � 19.



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.



Example 2A: Finding the GCF of Numbers

Find the GCF of each pair of numbers.

100 and 60

factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The GCF of 100 and 60 is 20.

List all the factors.

Circle the GCF.

Method 1 List the factors.



Example 2B: Finding the GCF of Numbers


26 and 52

26 = 2 � 13

52 = 2 � 2 � 13


Align the common factors.

2 � 13 = 26


Method 2 Prime factorization.



Check It Out! Example 2a


12 and 16

factors of 12: 1, 2, 3, 4, 6, 12

factors of 16: 1, 2, 4, 8, 16


List all the factors.

Circle the GCF.

Method 1 List the factors.



Check It Out! Example 2b


15 and 25

25 = 1 � 5 � 5



1 � 5 = 5

15 = 1 � 3 � 5

Method 2 Prime factorization.




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.



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 � x � x = 3x2

Write the prime factorization of

each coefficient and write

powers as products.


Find the product of the common

factors.

The GCF of 15x3 and 9x2 is 3x2.



Example 3B: Finding the GCF of Monomials


8x2 and 7y3

8x2 = 2 � 2 � 2 � x � x

7y3 = 7 � y � y � y

Write the prime

factorization of each

coefficient and write

powers as products.

Align the common

factors.

There are no

common factors

other than 1. The GCF 8x2 and 7y3 is 1.



If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power.

Helpful Hint



Check It Out! Example 3a


18g2 and 27g3

18g2 = 2 � 3 � 3 � g � g

27g3 = 3 � 3 � 3 � g � g � g

3 � 3 � g � g

The GCF of 18g2 and 27g3 is 9g2.

Write the prime factorization

of each coefficient and

write powers as products.


Find the product of the

common factors.



Check It Out! Example 3b


16a6 and 9b

9b = 3 � 3 � b

16a6 = 2 � 2 � 2 � 2 � a � a � a � a � a � a

Write the prime factorization of each coefficient and write powers as products.


There are no common factors other than 1.

The GCF of 16a6 and 9b is 1.



Check It Out! Example 3c


8x and 7v2

8x = 2 � 2 � 2 � x

7v2 = 7 � v � v

Write the prime factorization

of each coefficient and

write powers as products.


There are no common

factors other than 1. The GCF of 8x and 7v2 is 1.



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.



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 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.



18 chocolate milk cartons6 containers per row

= 3 rows

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.

Example 4 Continued




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.



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

Find the common

factors of 36

and 48.


Check It Out! Example 4 Continued

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.



36 pop CDs12 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.



Lesson Quiz: Part I


1. 50

2. 84


3. 18 and 75

4. 20 and 36

22 � 3 � 7

2 � 52

4

3



Lesson Quiz: Part II


5. 12x and 28x3

6. 27x2 and 45x3y2

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?

4x

17

9x2

lesson 7-1 factors and greatest common factors...holt mcdougal algebra 1 7-1 factors and greatest...

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