1. Objectives Write the prime factorization of numbers.
Find the GCF of monomials.

2. 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
1 12 
2 6
3 4

3. 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  98 49 7 1 2
98 = 
98 = 
The prime factorization of 98 is 2  7  7 or 2  72.

4. Example 2
Write the prime factorization of each number.
a. 40
b. 33 40  2 5 33 3 11
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.

5. Example 3
Write the prime factorization of each number.
c. 49
d. 19 49  19 1
49 = 7  7
19 = 1  19
The prime factorization of 49 is 7  7 or 72.
The prime factorization of 19 is 1  19.

6. 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.

7. Example 4: 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.

8. Example 5: Finding the GCF of Numbers
Find the GCF of each pair of numbers.
26 and 52
Method 2 Prime factorization.
Write the prime factorization of each number.
26 =  13
52 = 2  2  13
Align the common factors.
2  13 = 26
The GCF of 26 and 52 is 26.

9. Example 6
Find the GCF of each pair of numbers.
12 and 16
Method 1 List the factors.
List all the factors.
factors of 12: 1, 2, 3, 4, 6, 12
Circle the GCF.
factors of 16: 1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.

10. Example 7
Find the GCF of each pair of numbers.
15 and 25
Method 2 Prime factorization.
Write the prime factorization of each number.
15 = 1  3  5
25 = 1  5  5
Align the common factors.
1  = 5
The GCF of 15 and 25 is 5.

11. Example 8: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
15x3 and 9x2
Write the prime factorization of each coefficient and write powers as products.
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
Align the common factors.
3  x  x = 3x2
Find the product of the common factors.
The GCF of 15x3 and 9x2 is 3x2.

12. Example 9: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
8x2 and 7y3
Write the prime factorization of each coefficient and write powers as products.
8x2 = 2  2  2  x  x
7y3 =  y  y  y
Align the common factors.
There are no common factors other than 1.
The GCF 8x2 and 7y3 is 1.

13. Example 10
Find the GCF of each pair of monomials.
18g2 and 27g3
Write the prime factorization of each coefficient and write powers as products.
18g2 = 2  3  3  g  g
27g3 =  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.

14. Find the GCF of each pair of monomials.
Example 11
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 =  3  b
Align the common factors.
The GCF of 16a6 and 9b is 1.
There are no common factors other than 1.

15. Example 12
Find the GCF of each pair of monomials.
8x and 7v2
Write the prime factorization of each coefficient and write powers as products.
8x = 2  2  2  x
7v2 =  v  v
Align the common factors.
There are no common factors other than 1.
The GCF of 8x and 7v2 is 1.

16. Example 13: 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.

17. Example 13 Continued
Find the common factors of 18 and 24.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 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.

18. Example 13 Continued
18 chocolate milk cartons
6 containers per row
= 3 rows
24 regular milk cartons
= 4 rows
When the greatest possible number of types of milk is in each row, there are 7 rows in total.

19. Homework
Read section 7-1 in the workbook Workbook page 375: 1 – 10