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Name 7.1 ? Algebraic Reasoning—5.4.A MATHEMATICAL PROCESSES 5.1.C, 5.1.D, 5.1.F Factors and Divisibility Essential Question How can you tell whether one number is a factor of another number? Unlock Unlock the the Problem Problem Students in Carlo’s art class painted 32 square tiles for a mosaic. They will arrange the tiles to make a rectangle. Can the rectangle have 32 tiles arranged into 3 equal rows, without gaps or overlaps? A composite number is a whole number greater than 1 that has more than two factors. You can use models and division to find factors of composite numbers. One Way Draw a model. Think: Try to arrange the tiles into 3 equal rows to make a rectangle. A rectangle __ have 32 tiles arranged into 3 equal rows. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Artville/Getty Images Another Way Use division. If 3 is a factor of 32, then the unknown factor in 3 × ■ = 32 is a whole number. 3)‾ 3 2 ▲ Mosaics are decorative patterns made with pieces of glass or other materials. Math Idea A factor of a number divides the number evenly. This means the quotient is a whole number and the remainder is 0. Think: Divide to see whether the unknown factor is a whole number. Math Talk Mathematical Processes The unknown factor in 3 × ■ = 32 __ a whole number. So, a rectangle __ have 32 tiles arranged in 3 rows. Explain how the model relates to the quotient and remainder for 32 ÷ 3. Module 7 291 Divisibility Rules A number is divisible by another Divisibility Rules number if the quotient is a counting number and the remainder is 0. Some numbers have a divisibility rule. You can use a divisibility rule to tell whether one number is a factor of another. Is 6 a factor of 72? Think: If 72 is divisible by 6, then 6 is a factor of 72. Test for divisibility by 6: Is 72 even? _ Number Divisibility Rule 2 The number is even. 3 The sum of the digits is divisible by 3. 5 The last digit is 0 or 5. 6 The number is even and divisible by 3. 9 The sum of the digits is divisible by 9. What is the sum of the digits of 72? _+_=_ Is the sum of the digits divisible by 3? Math Talk _______ Mathematical Processes How are divisibility and factors related? Explain. 72 is divisible by _ . So, 6 is a factor of 72. Share Share and and Show Show 1. Is 4 a factor of 28? Draw a model to help. Think: Can you make a rectangle with 28 squares in 4 equal rows? Is 5 a factor of the number? Write yes or no. 2. 27 _ 292 3. 30 _ 4. 36 _ 5. 53 _ © Houghton Mifflin Harcourt Publishing Company 4 _ a factor of 28. Name Problem Problem Solving Solving List all the factor pairs in the table. Use a model or paper and pencil to help. 6. Factors of 24 7. Factors of 39 _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ List all the factor pairs for the number. Make a table to help. 8. 56 10. 9. 64 What’s the Error? George said if 2 and 4 are factors of a number, then 8 is a factor of the number. Is he correct? Explain. Problem Problem Solving Solving Use the table to solve 11–12. 11. Multi-Step Dirk bought a set of stamps. The number of stamps in the set he bought is divisible by 2, 3, 5, 6, and 9. Which set is it? © Houghton Mifflin Harcourt Publishing Company 12. Stamps Sets Country Number of stamps Germany 90 Sweden 78 Japan 63 Canada 25 Multi-Step Geri wants to put 6 stamps on some pages in her stamp book and 9 stamps on other pages. Explain how she could do this with the stamp set for Sweden. Module 7 • Lesson 1 293 Mathematical Processes M Model ¥ Reason ¥ Communicate M Daily Daily Assessment Assessment Task Task Fill in the bubble completely to show your answer. 13. There are 54 people attending a party. Carla is arranging chairs and tables. How should she arrange the chairs so that there will be an equal number of people seated at each table? A 12 chairs at each table B 8 chairs at each table C 9 chairs at each table D 10 chairs at each table 14. Analyze Jake organizes 48 marbles into packs. He places the same number of marbles into each pack. How could he arrange the marbles? A 8 or 10 in each pack B 9 or 12 in each pack C 10 or 12 in each pack D 8 or 12 in each pack 15. Multi-Step Megan has 34 rocks in her rock collection. She wants to put 5 rocks in some cases and 7 rocks in other cases. How could she arrange the rocks? A Have 6 cases of 5 rocks, and 1 case of 7 rocks. B Have 4 cases of 5 rocks, and 2 cases of 7 rocks. C Have 2 cases of 5 rocks, and 4 cases of 7 rocks. D Have 5 cases of 5 rocks, and 2 cases of 7 rocks. TEXAS Test Prep 16. Mrs. Mastrioni bought a set of 80 stamps. She wanted to give all the stamps 294 A 2 or 3 students. B 2 or 6 students. C 2, 4, 5, or 8 students. D 2, 4, 8, or 9 students. © Houghton Mifflin Harcourt Publishing Company to her students as a reward. She could give equal numbers of stamps to H o mewo rk and Practice 7.1 Algebraic Reasoning—5.4.A MATHEMATICAL PROCESSES 5.1.C, 5.1.D, 5.1.F Name Factors and Divisibility List all the factor pairs in the table. 1. Factors of 30 2. Factors of 15 _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ _ × _ = _ _, _ List all the factor pairs for the number. Make a table to help. 3. 18 4. 45 5. 36 6. 63 Problem Problem Solving Solving 7. Planners for the city’s botanical garden are © Houghton Mifflin Harcourt Publishing Company planting a rose garden. They want to plant 56 rose bushes. Can they arrange the rose bushes into 4 equal rows? Explain. 8. A marching band with 75 members makes two rectangular marching formations. One rectangle has 6 rows and the other rectangle has 9 rows. Explain how the members can form the two rectangles. Module 7 • Lesson 1 295 TEXAS Test Prep Lesson Lesson Check Check Fill in the bubble completely to show your answer. She places the same number of favors in each bag. How could she arrange the favors? A 60 B 33 A 3 or 6 in each bag C 46 B 6 or 9 in each bag D 21 C 7 or 8 in each bag D 7 or 9 in each bag 11. A package of marbles contains 20 blue 12. The fifth-grade students participating in the marbles, 38 red marbles, 48 green marbles, and 16 yellow marbles. Which color of marbles can be divided evenly between 6 people? math competition will be evenly divided into teams of 3 students each. How many students could be participating? A blue A 25 B red B 21 C green C 23 D yellow D 26 13. Multi-Step The librarian has 94 books to organize. He wants to put 5 books on some shelves and 6 books on other shelves. How could he arrange the books? A B 296 10. Jo Beth organizes 63 party favors into gift bags. Have 8 shelves of 5 books, and 9 shelves of 6 books. Have 9 shelves of 5 books, and 8 shelves of 6 books. C Have 5 shelves of 6 books, and 6 shelves of 5 books. D Have 7 shelves of 5 books, and 6 shelves of 6 books. 14. Multi-Step A group of 72 students went on a field trip. The students traveled in vans, with an equal number of girls and boys in each van. If 42 boys went on the trip, how many vans were there? A 9 B 8 C 7 D 6 © Houghton Mifflin Harcourt Publishing Company 9. Which number is divisible by 2, 3, and 6?