Factors and Greatest Common Factors 8 1 Common

Factors and Greatest Common Factors 8 -1 Common Factors Warm Up Lesson Presentation Lesson Quiz Holt Algebra 11

8 -1 Factors and Greatest Common Factors Warm Up Tell whether the second number is a factor of the first number 1. 50, 6 no 2. 105, 7 yes 3. List the factors of 28. ± 1, ± 2, ± 4, ± 7, ± 14, ± 28 Tell whether each number is prime or composite. If the number is composite, write it as the product of two numbers. 4. 11 prime Holt Algebra 1 5. 98 composite; 49 2

8 -1 Factors and Greatest Common Factors Objectives Write the prime factorization of numbers. Find the GCF of monomials. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Vocabulary prime factorization greatest common factor Holt Algebra 1

8 -1 Factors and Greatest Common Factors 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 Holt Algebra 1

8 -1 Factors and Greatest Common Factors 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 Holt Algebra 1

8 -1 Factors and Greatest Common Factors Remember! A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Example 1: Writing Prime Factorizations Write the prime factorization of 98. Method 1 Factor tree Method 2 Ladder diagram Choose any two factors Choose a prime factor 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 1. 98 2 98 7 49 2 49 7 7 1 98 = 2 7 7 The prime factorization of 98 is 2 7 7 or 2 72. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 1 Write the prime factorization of each number. a. 40 40 2 20 2 10 2 5 40 = 23 5 The prime factorization of 40 is 2 2 2 5 or 23 5. Holt Algebra 1 b. 33 11 33 3 33 = 3 11 The prime factorization of 33 is 3 11.

8 -1 Factors and Greatest Common Factors Check It Out! Example 1 Write the prime factorization of each number. c. 49 d. 19 49 7 7 49 = 7 7 The prime factorization of 49 is 7 7 or 72. Holt Algebra 1 1 19 19 19 = 1 19 The prime factorization of 19 is 1 19.

8 -1 Factors and Greatest Common 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. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Example 2 A: Finding the GCF of Numbers Find the GCF of each pair of numbers. 100 and 60 Method 1 List the factors of 100: 1, 2, 4, 5, 10, 25, 50, 100 List all the 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. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Example 2 B: Finding the GCF of Numbers Find the GCF of each pair of numbers. 26 and 52 Method 2 Prime factorization. 26 = 2 13 52 = 2 2 13 = 26 Write the prime factorization of each number. Align the common factors. The GCF of 26 and 52 is 26. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 2 a Find the GCF of each pair of numbers. 12 and 16 Method 1 List the factors of 12: 1, 2, 3, 4, 6, 12 List all the factors of 16: 1, 2, 4, 8, 16 Circle the GCF. The GCF of 12 and 16 is 4. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 2 b Find the GCF of each pair of numbers. 15 and 25 Method 2 Prime factorization. 15 = 1 3 5 25 = 1 5 5 1 Holt Algebra 1 5=5 Write the prime factorization of each number. Align the common factors.

8 -1 Factors and Greatest Common Factors 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. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Example 3 A: Finding the GCF of Monomials Find the GCF of each pair of monomials. 15 x 3 and 9 x 2 15 x 3 = 3 5 x x x 9 x 2 = 3 3 x x 3 Write the prime factorization of each coefficient and write powers as products. Align the common factors. x x = 3 x 2 Find the product of the common factors. The GCF of 15 x 3 and 9 x 2 is 3 x 2. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Example 3 B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8 x 2 and 7 y 3 Write the prime factorization of each 8 x 2 = 2 2 2 x x coefficient and write 7 y 3 = 7 y y y powers as products. Align the common factors. The GCF 8 x 2 and 7 y 3 is 1. Holt Algebra 1 There are no common factors other than 1.

8 -1 Factors and Greatest Common Factors Helpful Hint If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 3 a Find the GCF of each pair of monomials. 18 g 2 and 27 g 3 18 g 2 = 2 3 3 27 g 3 = g g Write the prime factorization of each coefficient and write powers as products. 3 3 3 g g g Align the common factors. 3 3 g g Find the product of the common factors. The GCF of 18 g 2 and 27 g 3 is 9 g 2. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 3 b Find the GCF of each pair of monomials. 16 a 6 and 9 b 16 a 6 = 2 2 a a a 9 b = The GCF of 16 a 6 and 9 b is 1. Holt Algebra 1 Write the prime factorization of each coefficient and write powers as products. 3 3 b Align the common factors. There are no common factors other than 1.

8 -1 Factors and Greatest Common Factors Check It Out! Example 3 c Find the GCF of each pair of monomials. 8 x and 7 v 2 8 x = 2 2 2 x 7 v 2 = 7 v v The GCF of 8 x and 7 v 2 is 1. Holt Algebra 1 Write the prime factorization of each coefficient and write powers as products. Align the common factors. There are no common factors other than 1.

8 -1 Factors and Greatest Common Factors 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. Holt Algebra 1

8 -1 Factors and Greatest Common Factors 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. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Example 4 Continued 18 chocolate milk cartons = 3 rows 6 containers per row 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. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 4 Adrianne is shopping for a CD storage unit. She has 36 CDs by pop music artists and 48 CDs by country music artists. She wants to put the same number of CDs on each shelf without putting pop music and country music CDs on the same shelf. If Adrianne puts the greatest possible number of CDs on each shelf, how many shelves does her storage unit need? The 36 pop and 48 country CDs must be divided into groups of equal size. The number of CDs in each row must be a common factor of 36 and 48. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Check It Out! Example 4 Continued Find the common Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 factors of 36 and 48. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 The GCF of 36 and 48 is 12. The greatest possible number of CDs on each shelf is 12. Find the number of shelves of each type of CDs when Adrianne puts the greatest number of CDs on each shelf. Holt Algebra 1

8 -1 Factors and Greatest Common Factors 36 pop CDs 12 CDs per shelf = 3 shelves 48 country CDs 12 CDs per shelf = 4 shelves When the greatest possible number of CD types are on each shelf, there are 7 shelves in total. Holt Algebra 1

8 -1 Factors and Greatest Common Factors Lesson Quiz: Part I Write the prime factorization of each number. 1. 50 2 52 2. 84 22 3 7 Find the GCF of each pair of numbers. 3. 18 and 75 3 4. 20 and 36 4 Holt Algebra 1

8 -1 Factors and Greatest Common Factors Lesson Quiz: Part II Find the GCF of each pair of monomials. 5. 12 x and 28 x 3 4 x 6. 27 x 2 and 45 x 3 y 2 9 x 2 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? 17 Holt Algebra 1