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

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 1 4 3 2 2 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. Page 1

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!

Example 1: Writing Prime Factorizations Page 2 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 2 49 7 7  98 49 7 1 2 98 = 2 7 7  98 = 2 7 7  The prime factorization of 98 is 2  7  7 or 2  72.

Write the prime factorization of 40. Your Turn 1 Page 2 Write the prime factorization of 40. 40 2 20 2 10  2 5 40 = 2  2  2  5 The prime factorization of 40 is 2  2  2  5 or 23  5.

The greatest of the common factors is 4. Page 1 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.

Example 2: Finding the GCF of Numbers Page 3 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.

Method 1 List the factors. Your Turn 2 Page 3 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.

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

Your Turn 3 Find the GCF of 18g2 and 27g3. 18g2 = 2  3  3  g  g Page 4 Find the GCF of 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.

rows of chocolate-milk cartons Page 5 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

rows of chocolate-milk cartons Page 5 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.

18 chocolate milk cartons 6 containers per row = 3 rows Example 4 Continued Page 5 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.

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