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Warm Up 1. 50, 6 2. 105, 7 3. List the factors of 28.

Warm Up 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. no. yes. ±1, ±2, ±4, ±7,. ±14, ±28. prime.

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Warm Up 1. 50, 6 2. 105, 7 3. List the factors of 28.

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  1. Warm Up 1. 50, 62. 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 no yes ±1, ±2, ±4, ±7, ±14, ±28 prime composite; 49  2 4. 11 5. 98

  2. 8-1 Factors and Greatest Common Factors Holt Algebra 1

  3. 1 12 3 4 1 4 3     2 6 2 2 3    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

  4. 1 12 3 4 1 4 3     2 6 2 2 3    The circled factorization is the prime factorization because all the factors are prime numbers. Factorizations of 12

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

  6. 40 33 11  2 20 3  2 10  2 5 Check It Out! Example 1 Write the prime factorization of each number. a. 40 b. 33 40 = 23 5 33 = 3  11 The prime factorization of 40 is 2  2  2  5 or 23 5. The prime factorization of 33 is 3  11.

  7. 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.

  8. 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.

  9. 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.

  10. Check It Out! Example 2b Find the GCF of each pair of numbers. 15 and 25 Method 2 Prime factorization. Write the prime factorization of each number. 15 = 1  3  5 25 = 1  5  5 Align the common factors. 1  5 = 5

  11. 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 Align the common factors. 9x2 = 3  3 x  x 3 x  x = 3x2 Find the product of the common factors. The GCF of 3x3 and 6x2 is 3x2.

  12. 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 7y is 1.

  13. 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.

  14. 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.

  15. 18 chocolate milk cartons = 3 rows 6 containers per row 24 regular milk cartons = 4 rows 6 containers per row Example 4 Continued When the greatest possible number of types of milk is in each row, there are 7 rows in total.

  16. Lesson Quiz: Part 1 Write the prime factorization of each number. 1. 50 2. 84 Find the GCF of each pair of numbers. 3. 18 and 75 4. 20 and 36 2  52 22 3  7 3 4

  17. Lesson Quiz: Part II Find the GCF each pair of monomials. 5. 12x and 28x3 6. 27x2 and 45x3y2 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? 4x 9x2 17

  18. Warm-Up Write the prime factorization of each number. 1. 50 2. 84 Find the GCF of each pair of numbers. 3. 18 and 75 4. 20 and 36 2  52 22 3  7 3 4

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