1. Lesson Objectives: I will be able to …
Write prime factorization of numbers
Find the greatest common factor of monomials
Language Objective: I will be able to …
Read, write, and listen about vocabulary, key
concepts, and examples

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
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.
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3. 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!

4. Example 1: Writing Prime Factorizations
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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.

5. Write the prime factorization of 40.
Your Turn 1
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Write the prime factorization of 40. 40  2 5
40 = 2  2  2  5
The prime factorization of 40 is 2  2  2  5 or 23  5.

6. The greatest of the common factors is 4.
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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 2: Finding the GCF of Numbers
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Example 2: Finding the GCF of Numbers
Find the GCF of 100 and 60.
Method 1 List the factors.
Method 2 Use prime
factorization.
factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
100 = 2  2  5  5
factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
60 = 2  2  3  5
2  2  5
= 20
The GCF of 100 and 60 is 20.

8. Method 1 List the factors.
Your Turn 2
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Find the GCF of 12 and 16.
Method 1 List the factors.
factors of 12: 1, 2, 3, 4, 6, 12
factors of 16: 1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.

9. Example 3: Finding the GCF of Monomials
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Find the GCF of 15x3 and -9x2.
15x3 =  5  x  x  x
-9x2 = -  3  3  x  x
3  x  x = 3x2
The GCF of 15x3 and -9x2 is 3x2.

10. Your Turn 3 Find the GCF of 18g2 and 27g3. 18g2 = 2  3  3  g  g
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Find the GCF of 18g2 and 27g3.
18g2 = 2  3  3  g  g
27g3 =  3  3  g  g  g
3  3  g  g
The GCF of 18g2 and 27g3 is 9g2.

11. rows of chocolate-milk cartons
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Example 4: School Application
A school cafeteria has 18 chocolate-milk cartons and 24 regular-milk cartons. The lunch server 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 lunch server puts the greatest possible number of cartons in each row?
# of rows = ? …
rows of chocolate-milk cartons
rows of regular-milk cartons

12. rows of chocolate-milk cartons
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Example 4 Continued
# of rows = ? …
rows of chocolate-milk cartons
rows of regular-milk cartons
The 18 chocolate and 24 regular milk cartons must be divided into groups of
6 cartons per row
equal size. The number of cartons in each row must be a common factor of 18 and 24.
Factors of 18: 1, 2, 3, 6, 9, 18
The GCF of 18 and 24 is 6.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 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.

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

14. Classwork Assignment #11
Holt 8-1 #17, 18, 21, 22, 25, 26, 29, 30, 32, 33, 38, 41, 42, 45, 47, 49, 57