1. Warm Up

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. 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.
Prime factorization: When factors of a product are all prime. None of the factors can be factored further.
Factorizations of 12
1 12 
2 6
3 4

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

5. Example 1: Writing Prime Factorizations
Write the prime factorization of 98.
Method 1 Factor tree
Choose any two factors
of 98 to begin. Keep finding factors until each branch ends in a prime factor. 98 
98 = 
The prime factorization of 98 is 2  7  7 or 2  72.

6. Check It Out! Example 1
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.

7. Check It Out! Example 1
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.

8. 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.

9. 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.

10. Example 2B: 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.

11. Check It Out! Example 2b
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

12. You can also find the GCF of monomials that include variables
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.

13. Example 3A: 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 3x3 and 6x2 is 3x2.

14. Example 3B: 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 7y is 1.

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

16. Check It Out! Example 3a
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.

17. Find the GCF of each pair of monomials.
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 =  3  b
Align the common factors.
The GCF of 16a6 and 7b is 1.
There are no common factors other than 1.

18. 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.

19. Example 4 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.

20. Example 4 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.

21. Lesson Quiz: Part 1
Write the prime factorization of each number.
1. 50
2. 84
Find the GCF of each pair of numbers.
3. 18 and 75
4. 20 and 36
2  52
22  3  7 3 4

22. Lesson 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 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
9x2 17