8 1 Factors and Greatest Common Factors Warm

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 Holt Algebra 11

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. Factorizations of 12 Holt Algebra 1

8 -1 Factors and Greatest Common Factors The circled factorization is the prime factorization because all the factors are prime numbers. Factorizations of 12 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 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 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 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 3 x 3 and 6 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 is 1. Holt Algebra 1 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 Lesson Quiz: Part 1 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 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

8 -1 Factors and Greatest Common Factors Warm-Up 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