8-1 Factors and Greatest Common Factors Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

8-1 Factors and Greatest Common Factors

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California Standards

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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


Preparation for 11.0 Students apply basic factoring techniques to second- and simple third degree-polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

California Standards


prime factorizationgreatest common factor

Vocabulary


The 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.

Remember that a prime number is a whole number that has exactly two positive factors, itself and 1. The number 1 is not prime because it has only one factor.


The order of factors does not change the product, but there is only one example 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

8-1 Factors and Greatest Common FactorsAdditional Example 1: Writing Prime Factorizations

Write the prime factorization of 98.

Method 1 Factor tree Method 2 Ladder diagramChoose any two factorsof 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

982 49

7 7

9849

71

2

77

98 = 2 7 7

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

8-1 Factors and Greatest Common FactorsCheck It Out! Example 1

Write the prime factorization of each number.

a. 40

402 20

2 10

2 5

333

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.


Check It Out! Example 1


c. 49 d. 19

49

7 71919

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.


Additional Example 2A: Finding the GCF of Numbers

Find the GCF of the 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.


Additional 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 = 5 5


Align the common factors.5

15 = 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.


Additional Example 3A: Finding the GCF of Monomials

Find the GCF of each pair of monomials.

15x3 and 9x2

15x3 = 3 5 x x x9x2 = 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.


Additional Example 3B: Finding the GCF of Monomials


8x2 and 7y3

8x2 = 2 2 2 x x

7y3 = 7 y y y



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.



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




The GCF of 16a6 and 9b is 1.

8-1 Factors and Greatest Common FactorsCheck It Out! Example 3c


8x and 7v2

8x = 2 2 2 x

7v2 = 7 v v




The GCF of 8x and 7v2 is 1.


Additional 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.


Additional 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.

Additional Example 4 Continued

8-1 Factors and Greatest Common FactorsCheck 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.


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 CD 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.

8-1 Factors and Greatest Common FactorsLesson Quiz: Part I


1. 50

2. 84

Find the GCF of each pair of numbers.

3. 18 and 75

4. 20 and 36

22 3 7

2 52

4

3

8-1 Factors and Greatest Common FactorsLesson Quiz: Part II

Find the GCF each pair of monomials.

5. 12x and 28x3

6. 27x2 and 45x3y2

7. Cindi is planting a rectangular flower bed with 40 orange flowers 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

Documents

8-1 Factors and Greatest Common Factors Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview