8-1 Factors and Greatest Common Factors

Transcription

1 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

2 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

3 Example 1: Writing Prime Factorizations tml?contentsrc=6365/6365.xml Write the prime factorization of 98. Method 1 Factor tree Method 2 Ladder diagram Choose any two factors Choose a prime factor of 98 of 98 to begin. Keep finding to begin. Keep dividing by factors until each branch prime factors until the ends in a prime factor. quotient is = = The prime factorization of 98 is or

4 Check It Out! Example 1 Write the prime factorization of each number. a. 40 b. 33

5 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.

6 Example 2A: Finding the GCF of Numbers html?contentsrc=7545/7545.xml 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, 6, 10, 12, 15, 20, 30, 60 List all the factors. Circle the GCF. The GCF of 100 and 60 is 20.

7 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. Align the common factors.

8 Example 3A: Finding the GCF of Monomials r.html?contentsrc=7546/7546.xml Find the GCF of each pair of monomials. 15x 3 and 9x 2 15x 3 = 3 5 x x x 9x 2 = 3 3 x x 3 x x = 3x 2 Write the prime factorization of each coefficient and write powers as products. Align the common factors. Find the product of the common factors. The GCF of 15x 3 and 9x 2 is 3x 2.

9 Example 3B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8x 2 and 7y 3

10 Example 4: Application r.html?contentsrc=7547/7547.xml 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.

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

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