7-1 FACTORS AND COMMON FACTORS CHAPTER 7

OBJECTIVES Write the prime factorization of numbers. Find the GCF of monomials.

FACTORIZATION 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       

PRIME FACTORIZATION 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. Factorizations of 12       

REMEMBER 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 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 =  

SOLUTION Method 2 Ladder diagram Choose a prime factor of 98 to begin. Keep dividing by prime factors until the quotient is =   The prime factorization of 98 is 2  7  7 or 2  7 2.

CHECK IT OUT! Write the prime factorization of each number. a. 40 The prime factorization of 40 is 2  2  2  5 or 2 3  5. b. 33 The prime factorization of 33 is 3  11. c. 49 The prime factorization of 49 is 7  7 or 7 2. d. 19 The prime factorization of 19 is 1  19.

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 factor of 12 and 32 is 4.

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 factors of 60: 1, 2, 3, 4, 5, Circle the GCF. 6, 10, 12, 15, 20, 30, 60 The GCF of 100 and 60 is 20.

EXAMPLE 2B: FINDING THE GCF OF NUMBERS Find the GCF of each pair of numbers. 26 and 52 Method 2 Prime factorization 26 = 2  = 2  2  13 2  13 = 26 The GCF of 26 and 52 is 26.

CHECK IT OUT! Find the GCF of each pair of numbers A) 12 and 16 Ans: The GCF of 12 and 16 is 4 B) 15 and 25 Ans:The GCF of 15 and 25 is 5.

GCF IN MONOMIALS 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.

EXAMPLE 3A: FINDING THE GCF OF MONOMIALS Find the GCF of each pair of monomials. 15x 3 and 9x 2 Sol: 15x 3 = 3  5  x  x  x 9x 2 = 3  3  x  x 3  x  x = 3x 2 The GCF of 15x 3 and 9x 2 is 3x 2.

EXAMPLE Find the GCF of each pair of monomials. 8x 2 and 7y 3 Sol: 8x 2 = 2  2  2  x  x 7y 3 = 7  y  y  y The GCF 8x 2 and 7y 3 is 1. There are no common factors other than 1.

CHECK IT OUT!! Find the GCF of each pair of monomials. A) 18g 2 and 27g 3 Answer: The GCF of 18g 2 and 27g 3 is 9g 2. B) 16a 6 and 9b Answer: GCF is 1

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?

SOLUTION 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 24 regular milk cartons 6 containers per row = 4 rows When the greatest possible number of types of milk is in each row, there are 7 rows in total.

STUDENT GUIDED PRACTICE Do even problems from 2-15 in your book page 459

HOMEWORK Do even problems in your book page 459

CLOSURE Today we learned about prime factorization and the greatest common factor Next class we are going to keep learning about factorization