FACTORS AND GREATEST COMMON FACTORS

The numbers that are multiplied to find a product are called factors of that product. A number is evenly 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 : 

1 ⋅ 12

2 ⋅ 6

3 ⋅ 4

1 ⋅ 4 ⋅ 3

2 ⋅ 2 ⋅ 3

The order of the factors does not change the product, but there is only one example above that cannot be factored further. The factorization 2 ⋅ 2 ⋅ 3 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.

Writing Prime Factorizations

Example 1 : 

Write the prime factorization of 60.

Solution : 

Method 1 : Factor tree

Choose any two factors of 60 to begin. Keep finding factors until each branch ends in a prime factor. 

Method 2 : Ladder diagram

Choose a prime factor of 60 to begin. Keep dividing by prime factors until the quotient is 1.

Greatest Common Factor

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.

Finding the GCF of Numbers

Find the GCF of each pair of numbers.

Example 2 : 

24 and 60

Solution : 

Method 1 : List the factors

Factors of 24 : 1, 2, 3, 4, 6, 8, 12, 24 

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

The GCF of 24 and 60 is 12. 

Example 3 : 

18 and 27

Solution :

Method 2 : Prime factorization

18 : 2 ⋅ 3 ⋅ 3

27 : 3 ⋅ 3 ⋅ 3

3 ⋅ 3  =  9

The GCF of 18 and 27 is 9. 

Finding the GCF of Monomials

Find the GCF of each pair of monomials.

Example 4 : 

3x³ and 6x²

Solution : 

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

3x³  =  2 ⋅ 3 ⋅ x ⋅ x ⋅ x

3x²  =  2 ⋅ 3 ⋅ x ⋅ x

Find the product of the common factors.

3 ⋅ x ⋅ x  =  3x²

The GCF of 3x³ and 6x² is 3x².

Example 5 : 

4x² and 5y³

Solution : 

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

4x²  =  2 ⋅ 2 ⋅ x ⋅ x

5y³  =  5 ⋅ y ⋅ y ⋅ y

There are no common factors other than 1.

The GCF of 4x² and 5y³ is 1. 

Technology Application

Example 6 : 

Joseph is creating a Web page that offers electronic greeting cards. He has 24 special occasion designs and 42 birthday designs. The cards will be displayed with the same number of designs in each row. Special occasion and birthday designs will not appear in the same row. How many rows will there be if Joseph puts the greatest possible number of designs in each row?

Solution :

The 24 special occasion designs and 42 birthday designs must be divided into groups of equal size. The number of designs in each row must be a common factor of 24 and 42.

Factors of 24 : 1, 2, 3, 4, 6, 8, 12, 24 

Factors of 42 : 1, 2, 3, 6, 7, 14, 21, 42

The GCF of 24 and 42 is 6.

The greatest possible number of designs in each row is 6. Find the number of rows of each group of designs when there are 6 designs in each row.

24 special occasion designs / 6 designs per row  =  4 rows

42 birthday designs / 6 designs per row  =  7 rows

When the greatest possible number of designs is in each row, there are 11 rows in total.

Example 7 :

Kiara baked 30 oatmeal cookies and 48 chocolate chip cookies to package in plastic containers for her friends at school. She wants to divide the cookies into identical containers so that each container has the same number of each kind of cookie. If she wants each container to have the greatest number of cookies possible, how many plastic containers does she need?

Solution :

Number of oatmeal cookies she made = 30

Number of chocolate chip cookies = 48

To pack this into the containers with same number of oatmeal cookies and chocolate chip cookies, we have to find the greatest common factor.

greatest common factor = 2 x 3

= 6

She can pack 6 cookies in each category,

Number of containers she need to pack oatmeal cookies 

= 30/6

= 5

Number of containers she need to pack chocolate chip cookies 

= 48/6

= 8

Total number of containers = 5 + 6

= 11 containers

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