Factors and Greatest 7-1 Common Factors Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Warm Up 1. 50, 6 2. 105, 7 3. List the factors of 28. 4. 11 5. 98 1. 50, 6 2. 105, 7 3. List the factors of 28. Tell whether each number is prime or composite. If the number is composite, write it as the product of two numbers. Tell whether the second number is a factor of the first number 4. 11 5. 98

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

Factors: numbers that are multiplied to find a product. *A number is divisible by its factors. Example: 12 can be factored several ways. How?? Factorizations of 12 1 12  2 6 3 4 1 4 3 2 2 3

Prime factorization: factoring a number into only prime numbers. *Only one way to write the prime factorization of a number. Factorizations of 12 1 12  2 6 3 4 1 4 3 2 2 3

Example 1: Writing Prime Factorizations Write the prime factorization of 98 and 25. Factor tree a. 98 b. 25 Choose any two factors of 98 to begin. Keep finding factors until each branch ends in a prime factor. 25 5 5  98 2 49 7 7  25 = 5  5 The prime factorization of 25 is 5  5 or 52. 98 = 2 7 7  The prime factorization of 98 is 2  7  7 or 2  72.

Check It Out! Example 1 Write the prime factorization of each number. a. 40 b. 33 c. 19

Common Factors: factors shared by two or more whole numbers. Greatest common factor (GCF): largest common factor between two or more numbers 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 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 List all the factors. 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.

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. 26 = 2  13 52 = 2  2  13 Align the common factors. 2  13 = 26 The GCF of 26 and 52 is 26.

Check It Out! Example 2a Find the GCF of each pair of numbers. 12 and 16 Method 1 List the factors.

Check It Out! Example 2b Find the GCF of each pair of numbers. 15 and 25 Method 2 Prime factorization.

Steps to find the GCF of monomials with variables Write the prime factorization of each coefficient Write all powers of variables as products. Find the product of the common factors.

Example 3A: Finding the GCF of Monomials Find the GCF of each pair of monomials. 15x3 and 9x2 Write the prime factorization of each coefficient and write powers as products. 15x3 = 3  5  x  x  x 9x2 = 3  3  x  x Align the common factors. 3  x  x = 3x2 Find the product of the common factors. The GCF of 15x3 and 9x2 is 3x2.

Example 3B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8x2 and 7y3 Write the prime factorization of each coefficient and write powers as products. 8x2 = 2  2  2  x  x 7y3 = 7  y  y  y Align the common factors. There are no common factors other than 1. The GCF 8x2 and 7y3 is 1.

Check It Out! Example 3a Find the GCF of each pair of monomials. 18g2 and 27g3 16a6 and 9b

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.

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

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

#18-24(evens), 25-36(all), 40, 41, 57, and 58, skip 30 HOMEWORK PG. 459-461 #18-24(evens), 25-36(all), 40, 41, 57, and 58, skip 30