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Primetime Short Answer List all the factors for the number. 1. 24 2. 31 3. List all the proper factors of 35. 4. Which number(s) 24, 31, 35 are prime numbers? Explain why. 5. Which number(s) 24, 31, 35 are composite numbers? Explain why. 6. a. You are playing the Product Game on a game board like the one shown. One of the paper clips is on 7. What products can you make by moving the other paper clip? b. List two multiples of 7 that are not on the game board. 7. Why isn’t the number 13 on the Product Game board? 8. Which of these numbers are square numbers? Explain. 25 36 48 9. a. List the factors of 16 and the factors of 28. b. Complete the Venn diagram. c. What is the greatest common factor of 16 and 28? 10. a. List the first five multiples of 15 and the first five multiples of 12. b. Complete the Venn diagram. c. What is the least common multiple of 15 and 12? d. Find a common multiple of 15 and 12 that is not in your lists. 11. Jill says 6 is a common factor of 56 and 36. Is she correct? Explain your reasoning. 12. Evonne and Dolphus found a new Product Game board. Three of the factors and one of the products were not filled in. a. What are the other three factors you would need in order to play the game using this board? b. What product is missing? 13. Terrapin Crafts wants to rent between 35 and 40 square yards of space for a big crafts show. The space must be rectangular, and the side lengths must be whole numbers. Find the number(s) between 35 and 40 with the most factor pairs that gives the greatest number of rectangular arrangements to choose from. 14. Two radio stations are playing the #1 hit song “2 Nice 2 B True” by Anita and the Goody-2Shoes. WMTH plays the song every 18 minutes. WMSU plays the song every 24 minutes. Both stations play the song at 3:00 P.M. When is the next time the stations will play the song at the same time? 15. Judith is planning a party for her younger brother. She has 36 prizes and 24 balloons. How many children can she have at the party so that each child gets an equal number of prizes and an equal number of balloons? Explain your answer. 16. Find three different ways to show factorizations (strings of factors) of the number 16. Do not use 1 as a factor. 17. Find the prime factorization of the following two numbers. Show your work. a. 72 b. 132 18. reasoning. 19. A number that is less than 85 has 26 and 6 as factors. Find the number and explain your What number has the prime factorization ? Show how you found the number. 20. Find the dimensions of all of the rectangles that can be made from 48 square tiles. Explain how you found your answers. 21. “Sam” and “Martha” are the local names for two lighthouses that guard a particularly dangerous part of the coast. Sam blinks every 12 seconds and Martha blinks every 8 seconds. They blink together at midnight. How many seconds will pass before they blink together again? 22. Carlos is packing sacks for treats at Halloween. Each sack has to have exactly the same stuff in it or the neighborhood kids complain. He has on hand 96 small candy bars and 64 small popcorn balls. a. What is the greatest number of treat sacks he can make? b. How many of each kind of treat is in one sack? 23. a. What is the greatest common factor of 30 and 42? b. Give a different common factor of 30 and 42. c. What is the least common multiple of 30 and 42? d. Give an additional common multiple of 30 and 42. 24. Dawson wrote the factorization . Without finding the actual number, how can Dawson tell if the number is even or odd? 25. Scarlett and Rhett were playing the Factor Game when Ashley looked over and saw that the numbers 1 to 15 were all circled. Ashley immediately said, “Oh, I see that your game is over.” Is Ashley correct? Explain. Describe how you can tell whether a given number is a multiple of the number shown. 26. 2 27. 3 28. 5 29. List all multiples of 6 between 1 and 100. What do these numbers have in common? 30. Mr. Matsumoto said, “I am thinking of a number. I know that to be sure I find all of the factor pairs of this number, I have to check all the numbers from 1 through 15.” a. What is the smallest number he could be thinking of? Explain your answer. b. What is the greatest number he could be thinking of? Explain your answer. 31. What is the mystery number? Clue 1 My number is between the square numbers 1 and 25. Clue 2 My number has exactly two factors. Clue 3 Both 66 and 605 are multiples of my number. 32. Use concepts you have learned in this unit to make a mystery number question. Each clue must contain at least one word from your vocabulary list. 33. a. List the first ten square numbers. b. Give all the factors for each number you listed in part (a). c. Which of the square numbers you listed have only three factors? d. If you continued your list, what would be the next square number with only three factors? 34. A mystery number is greater than 50 and less than 100. You can make exactly five different rectangles with the mystery number of tiles. Its prime factorization consists of only one prime number. What is the number? 35. A number has 4 and 5 as factors. a. What other numbers must be factors? Explain. b. What is the smallest the number could be? 36. Chairs for a meeting are arranged in six rows. Each row has the same number of chairs. a. What is the minimum possible number of chairs that could be in the room? b. Suppose 100 is the maximum number of chairs allowed in the meeting room. What other numbers of chairs are possible? 37. Gloomy Toothpaste comes in two sizes: 9 ounces for $0.89 and 12 ounces for $1.15. a. Ben and Aaron bought the same amount of toothpaste. Ben bought only 9-ounce tubes, and Aaron bought only 12-ounce tubes. What is the smallest possible number of tubes each boy bought? (Hint: Use your knowledge of multiples to help you.) b. Which size tube is the better buy? 38. Circle the letter(s) of the statements that are always true about any prime number. a. It is divisible by only itself and 1. b. It is a factor of 1. c. It is divisible by another prime number. d. It is always an odd number. 39. Tyrone claims that the longest string of factors for 48 is 48 = . Ian says there is a longer string. He wrote 48 = . Who is correct? Why? 40. What is the smallest number divisible by the first three prime numbers and the first three composite numbers? Explain. 41. Suppose you are playing the Factor Game on the 30-board. Your opponent goes first and chooses 29, giving you only 1 point. It is now your turn to choose a number. Which number would be your best move? Why? 42. Suppose the person who sits next to you was absent the day you played the Factor Game. On the back of this paper, write a note to him or her explaining the strategies you have discovered for winning the Factor Game. Include a description of how you decide which move to make when it is your turn. 43. Vicente made three dozen cookies for the student council bake sale. He wants to package them in small bags with the same number of cookies in each bag. a. List all the ways Vicente can package the cookies. b. If you were Vicente, how many cookies would you put in each bag? Why? c. Vicente spent $5.40 on ingredients for the cookies. The student council will pay him back for the money he spent. For each of the answers in part (a), determine how much the student council should charge for each bag of cookies so they make a profit yet still get students to buy the cookies. 44. Marcia has developed a rule for generating a number sequence. The first 6 numbers in her sequence are 7, 21, 42, 126, 252, 756. a. What is Marcia’s rule for finding the numbers in her sequence? Explain. b. What are the next two numbers in Marcia’s sequence? c. What is the greatest common factor (GCF) of all the terms in Marcia’s sequence? Explain your reasoning. 45. a. List two pairs of numbers whose least common multiple (LCM) is the same as their product. For example, the least common multiple of 5 and 6 is 30 and 5 × 6 = 30. b. List two pairs of numbers whose least common multiple is smaller than their product. For example, the least common multiple of 6 and 9 is 18 and 18 is less than 6 × 9. c. For a given pair of numbers, how can you tell whether the least common multiple will be less than or equal to their product? 46. a. Write the prime factorization of 900. b. From information in the prime factorization of 900, write five sentences about the number 900. Use vocabulary from the unit in each sentence. 47. For each of the following, use the set of clues to determine the secret number. a. Clue 1 The number has two digits. Clue 2 The number has 13 as a factor. Clue 3 The sum of the digits of the number is 11. b. Clue 1 The number is prime. Clue 2 The number is less than 19. Clue 3 The sum of the digits of the number is greater than 7. 48. The numbers 10, 20, and 30 on the 30-board in the Factor Game all have 10 as a factor. Does any number that has 10 as a factor also have 5 as a factor? Explain your reasoning. 49. The numbers 14, 28, and 42 on the 49-board in the Factor Game all have 7 as a factor and also have 2 as a factor. Does any number that has 7 as a factor also have 2 as a factor? Explain your reasoning. 50. Look carefully at the numbers 1–30 on the 30-board used for playing the Factor Game. Pick the two different numbers on the 30-board that will give you the largest number when you multiply them together, and then answer the following questions. a. What two numbers did you pick? What is the product of the two numbers? b. Explain why the product of the two numbers you chose is the largest product you can get using two different numbers from the 30-board. c. List all the proper factors of the product. Explain how you found the factors. 51. For each of the following, find three different numbers that can be multiplied together so that the given number is the product. Do not use 1 as one of the numbers. a. 150 b. 1,000 c. 24 d. 66 52. The number sequence 4, 6, 10 is a multiple of the number sequence 2, 3, 5 because the sequence 4, 6, 10 can be found by multiplying all the numbers in the sequence 2, 3, 5 by 2. That is, 4 = , 6 = , 10 = . a. The number sequence 15, 25, 10 is a multiple of what number sequence? b. Find two different sequences that are multiples of the number sequence 1, 4, 7. c. Given a number sequence, how many different sets of multiples of that sequence do you think there are? Explain your reasoning. 53. Given the following sets of numbers, write as many different multiplication and division statements as you can. For example, if the numbers are 5, 7, 35, you can write: a. 6, 4, 24 b. 96, 12, 8, 3, 32 c. 6, 27, 108, 12, 4, 18, 9 d. When is a number called a factor of a number? A divisor of a number? 54. Alicia has made a rectangle using 24 square tiles. If she adds the length and width of her rectangle together, she gets 11. What is the length and width of Alicia’s rectangle? Explain your reasoning. 55. Jennifer has made a rectangle using 48 square tiles. If she adds the length and width of her rectangle together she gets a prime number. What is the length and width of Jennifer’s rectangle? Explain your reasoning. 56. List all of the factor pairs for each of the following numbers. a. 56 b. 42 c. 31 d. 80 e. 75 f. 108 g. 225 57. Phillip is thinking of a number that is less than 20 and has three factor pairs. Phillip also says that if he adds together the factors in the factor pairs he gets 19, 11, and 9. What is Phillip’s number? Explain how you found your answer. 58. In each of the rectangles shown below, only the tiles along the length and width are shown. For each rectangle, explain how many square tiles it would take to make each rectangle. a. b. c. 59. a. Draw and label a Venn diagram in which one circle represents the factors of 12 and another circle represents the factors of 13. Place the numbers from 1 to 15 in the appropriate regions of the diagram. b. What do you notice about the numbers in the intersection? Why does this happen? c. What is another set of labels, one for each of the two circles, that gives the same numbers in the intersection as you found in part (b)? Explain your reasoning. 60. a. Draw and label a Venn diagram in which one circle represents the multiples of 5 and another circle represents the multiples of 2. Place the numbers from 1 to 40 in the appropriate regions of the diagram. b. What do you notice about the numbers in the intersection? Why does this happen? c. Where would you place 75 in the diagram? Where would you place 90? Explain your reasoning. 61. Karl added four numbers together and got an even sum. Three of the numbers are 42, 35, and 77. What can you say about the fourth number? Explain your reasoning. 62. On Saturdays, the #14 bus makes roundtrips between Susan’s school and the mall, and the #11 bus makes roundtrips between the mall and the museum. Next Saturday, Susan wants to take the bus from her school to the museum. A #14 bus leaves Susan’s school every 15 minutes, beginning at 7 A.M. It takes the bus 30 minutes to travel between the school and the mall. A #11 bus leaves the mall every 12 minutes, beginning at 7 A.M. a. If Susan gets on the #14 at 9:30 A.M., how long will she have to wait at the mall for a #11 bus? Explain your reasoning. b. If Susan gets on the #11 bus at the museum and arrives at the mall at 11:48 A.M., how long will she have to wait for the #14 bus? Explain your reasoning. c. At what times between 9 A.M. and noon are the #14 and #11 buses at the mall at the same time? Explain your reasoning. 63. Kyong has built two rectangles. Each has a width of 7 tiles. a. If each rectangle is made with an even number of tiles that is greater than 40 but less than 60, how many tiles does it take to make each rectangle? Explain your reasoning. b. What is the length of each of Kyong’s rectangles? Explain your reasoning. c. Without changing the number of tiles used to make either rectangle, Kyong rearranges the tiles of each rectangle into different rectangles. What is a possibility for the length and width of each of Kyong’s new rectangles? Explain your reasoning. 64. Jack plays on a basketball team after school (or on the weekend) every third day of the month. He babysits his younger brother after school every seventh day of the month. How many times during a 30-day month, if any, will Jack have a conflict between basketball and babysitting? Explain your reasoning. 65. Suppose you have two different numbers which are both prime. a. What is the least common multiple of the numbers? Explain your reasoning. b. What is the greatest common factor? Explain your reasoning. 66. Find the least common multiple and the greatest common factor for each pair of numbers: a. 8 and 12 b. 7 and 15 c. 11 and 17 d. 36 and 108 e. For which pairs in parts (a) – (d) is the least common multiple the product of the two numbers? Why is this so? What is special about the numbers in these pairs? 67. Find the greatest common factor of each pair of numbers: a. 4 and 12 b. 5 and 15 c. 10 and 40 d. 25 and 75 e. When is the greatest common factor of two numbers one of the two numbers? Explain your reasoning. 68. Find the prime factorization for each of the numbers below. a. 630 b. 144 c. 1,011 d. 133 e. 23 69. Solve each of the multiplication mazes given below. Record your solution for each maze by copying the maze on your paper and then tracing out the path through the maze. a. b. c. d. 70. For each of the pairs of numbers given below, find the greatest common factor and the least common multiple. a. 25 and 105 b. 27 and 81 c. 36 and 63 71. An odd number that is less than 160 has exactly three different prime factors. What is the number? Explain your reasoning. 72. What number has the prime factorization ? 73. a. Name a pair of numbers whose greatest common factor is the same as one of the numbers. b. Name another pair of numbers whose greatest common factor is the same as one of the numbers. c. Make a conjecture about what must be true about the least common multiple of any number pairs in which one number is the greatest common factor of the other number. 74. a. Are 45 and 64 relatively prime? Explain your reasoning. b. Are 25 and 36 relatively prime? Explain your reasoning. c. Is it possible for two numbers that are both even to be relatively prime? Why or why not? d. How can you choose one number so that it will be relatively prime to any other number? 75. Find all of the numbers less than 1,000 that have 3 as their only prime factor. Explain your strategy for finding all of these numbers. 76. A number sequence is an ordered series of numbers that follow a pattern or rule. Jason has developed a secret rule for generating his own number sequence. Here are the first five terms in the sequence: 3, 15, 45, 225, 675, ... and so on. Use Jason’s sequence to answer the following questions. a. What is Jason’s rule for finding the numbers in his number sequence? Explain how you found your answer. b. What are the next two terms in Jason’s number sequence? c. What is the greatest common factor of all the terms in Jason’s sequence, no matter how many new numbers he adds to the sequence? Explain your reasoning. 77. In the 1,000-locker problem, which students touched the lockers indicated? a. both lockers 13 and 19 b. lockers 12, 16, and 20 78. In the 1,000-locker problem, what was the last locker touched by the students indicated? a. both students 20 and 25 b. both students 13 and 19 c. all three students 3, 4, and 5 d. all three students 30, 40, and 50 79. A set of consecutive numbers that contains no prime numbers is called a prime desert. For example, the set {14, 15, 16} is a prime desert because it is a set of consecutive numbers and none of the numbers are prime. Find the prime desert that has the most numbers in it where all the numbers are less than 50. 80. For each of the sets of clues below, find the secret number. a. Clue 1 The number is less than 130. Clue 2 The number ends in a 5. Clue 3 The number is a multiple of a prime that is greater than 20, but less than 30. b. Clue 1 The number ends in a 0. Clue 2 The number is a multiple of 21. Clue 3 The number is less than 400. 81. Find the prime factorization of each of the following numbers. a. 190 b. 319 c. 255 d. 406 82. Test 71,094 for divisibility by 5. 83. Tell whether 5,136 is divisible by 2, 3, 4, 5, 9, or 10. Write none of these if applicable. 84. Write the prime factorization of 160. Use exponents where possible. 85. Mr. Wolfe wants to divide a 6th grade class of 40 students into equal groups. He wants the number of students in each group to be a prime number. a. In how many different ways can Mr. Tindell divide the class? b. How many students will be in each group? Multiple Choice Identify the choice that best completes the statement or answers the question. ____ a. 2 86. What is the greatest common factor of 16 and 28? b. 3 c. 4 d. 112 ____ a. 8 b. 11 c. 14 d. 24 87. Which of the following is a factor of 44? ____ a. 1 b. 5 c. 10 d. 30 88. Which number is a multiple of 15? ____ a. 2 b. 3 c. 4 d. 6 89. How many different factors does 20 have? ____ a. 11 b. 14 c. 24 d. 56 90. Which number is a common multiple of 7 and 4? ____ a. 91. Find the prime factorization of 160 ____ a. 2 b. 9 c. 252 d. 504 92. What is the least common multiple of 18 and 28? ____ a. 93. Which answer is always odd? c. b. d. b. odd + odd c. d. even + even e. ____ a. 94. Which string of factors below is not a factorization of 180 95. What is the least number that has 2, 3, and 4 as factors? c. b. d. ____ a. 6 b. 9 c. 12 d. 18 ____ 96. a. 28 by 7 c. 43 by 9 b. 43 by 4 d. 67 by 8 In which case is the first number divisible by the second? Use mental math. ____ 97. Test 72,238 for divisibility by 2, 5, or 10. a. It is divisible by 2, but not by 5 or 10. b. It is divisible by 5 and 10 but not by 2. c. It is divisible by 2, 5, and 10. d. It is divisible by 5, but not by 2 or 10. Find the LCM of the numbers. ____ a. 32 c. 16 b. 64 d. 48 98. 4, 16 Find the prime factorization of the number. ____ a. 99. 168 c. b. d. ____ a. 100. 360 ____ a. 45 b. 58 c. 58 d. 108 101. Which number is divisible by 5? ____ a. 109 b. 135 c. 116 d. 125 102. Which number is divisible by 3? ____ a. 7 b. 0 c. 6 d. 103. Find the missing digit to make 27,47_ divisible by 9. c. b. d. 3 ____ 977 428 2,552 1,800 104. Which number is divisible by 9 and 3? a. 428 b. 2,552 c. 1,800 d. 977 ____ 105. 2,447 and 5 2,227 and 9 4,366 and 10 1,644 and 2 For which pair of numbers is the first number divisible by the second number? a. 2,447 and 5 c. 2,227 and 9 b. 4,366 and 10 d. 1,644 and 2 ____ 1,272 3,786 8,891 9,634 a. 8,891 b. 9,634 c. 3,786 d. 1,272 106. Which number is divisible by 4? ____ a. 42 b. 3 c. 84 d. 126 107. Find the LCM of 6 and 21. ____ a. 53 b. 56 c. 57 d. 51 108. Tell which number is prime: 51, 53, 56, 57. ____ a. 4 b. 3 c. 12 d. 6 109. Find the GCF of 66 and 87. ____ 110. The local reader’s club has a set of 28 hardback books and a set of 44 paperbacks. Each set can be divided equally among the club members. What is the greatest possible number of club members? a. 308 b. 4 c. 2 d. 8 ____ a. 32 c. 111. List the factors to find the GCF of 32 and 56. 224 b. 24 d. 8 ____ a. 3 c. 2 b. 4 d. 1 112. Use a division ladder to find the GCF of 22 and 50. Find the GCF of the numbers. ____ a. 11 c. 8 b. 2 d. 1,408 113. 64, 16, 176 ____ a. 68 c. 8 b. 4 d. 240 114. 20, 48 ____ 115. Alejandro and Jean are distributing erasers and pencils to the art class. There are 40 erasers and 25 pencils. Each student receives the same number of pencils and the same number of erasers, and no supplies are left over. What is the greatest number of students in the class? a. 10 students c. 65 students b. 200 students d. 5 students List all the factors for the number. ____ 116. 60 a. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 c. 1, 2, 3, 4, 7, 14, 15, 20, 30, 60 b. 1, 2, 4, 5, 10, 20 d. 1, 2, 3, 4, 5, 10 ____ 117. Which number is composite? 43, 23, 17, 52, 67 a. 52 b. 23 c. 67 d. 43 Other 118. Is 3665 divisible by 9? How do you know? 119. Jennifer and Melissa are both moving and must pack their collections of CDs. Jennifer has 9 boxes and 108 CDs, while Melissa has 5 boxes and 193 CDs. Can each of them divide their CDs evenly among their boxes? Explain. Primetime Answer Section SHORT ANSWER 1. ANS: 1, 2, 3, 4, 6, 8, 12, 24 PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors factor | proper factors | whole number factors | whole number divisors 2. ANS: 1, 31 PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors 3. 1, 5, 7, 35 PTS: OBJ: STA: KEY: ANS: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors 4. ANS: 31; It has exactly two factors, one and the number itself. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers prime number | composite number | abundant numbers | perfect numbers | deficient numbers 5. ANS: 24, 35; These numbers have more than two factors. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers prime number | composite number | abundant numbers | perfect numbers | deficient numbers 6. ANS: a. 7, 14, 21, 28, 35, 49, 56 b. Possible answers: 42, 63 PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.3 Finding Multiples multiple | square numbers | divisible by | near-perfect numbers 7. ANS: The factors of 13 are 1 and 13. Both of these factors are not listed below the game board. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Check-Up 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.3 Finding Multiples multiple | square numbers | divisible by | near-perfect numbers 8. ANS: 25 and 36; Each of these numbers can be expressed as a number times itself. 5 × 5 = 25 and 6 × 6 = 36. If you made a tile model you could arrange each number of tiles into a square. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Check-Up 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor 9. ANS: a. Factors of 16: 1, 2, 4, 8, 16 Factors of 28: 1, 2, 4, 7, 14, 28 b. c. 4 PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Check-Up 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.3 Classifying Numbers KEY: Venn diagram | intersection | factor 10. ANS: a. Multiples of 15: 15, 30, 45, 60, 75 Multiples of 12, 24, 36, 48, 60; b. c. 60 d. Possible answers: 120, 180, 240, 300 PTS: 1 DIF: L2 REF: Prime Time | Check-Up 2 OBJ: Investigation 2: Whole-Number Patterns and Relationships NAT: NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B TOP: Problem 2.3 Classifying Numbers KEY: Venn diagram | intersection | multiple 11. ANS: Jill is incorrect. In order for 6 to be a common factor, both numbers must be divisible by 6. The number 56 cannot be divided evenly by 6. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Check-Up 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.3 Classifying Numbers KEY: Venn diagram | intersection | factor | common factors 12. a. 3, 4, 6 b. 81 PTS: OBJ: STA: KEY: ANS: 1 DIF: L2 REF: Prime Time | Partner Quiz Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.3 Finding Multiples multiple | square numbers | divisible by | near-perfect numbers 13. ANS: 36; If we consider an rectangle to be different from a rectangle, of the numbers 36, 37, 38, and 39, the number 36 gives nine choices of rectangles. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Partner Quiz Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions 14. ANS: In 72 minutes, or at 4:12 P.M. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Partner Quiz Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.1 Finding Common Multiples | Problem 3.2 Finding the Least Common Multiple common multiples | least common multiple | common factors | greatest common factor 15. ANS: Judith could have 1, 2, 3, 4, 6, or 12 children at the party. These numbers are common factors of 24 and 36. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Partner Quiz Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.3 Finding Common Factors | Problem 3.4 Finding the Greatest Common Factor common factors | common multiples | greatest common factor | least common multiple 16. ANS: Possible answers: ; ; PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.1 Finding Factor Strings prime factorization | factor tree | factoring 17. ANS: a. 72: b. 132: PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Finding the Longest Factor String factor tree | exponent | Fundamental Theorem of Arithmetic 18. ANS: The number is 78 = . PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Finding the Longest Factor String | Problem 4.3 Using Prime Factorizations factor tree | exponent | Fundamental Theorem of Arithmetic | relatively prime 19. ANS: The number is 2,100. = = 2,100 PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Finding the Longest Factor String factor tree | exponent | Fundamental Theorem of Arithmetic 20. ANS: , , , , . List all the factor pairs, starting with 1 and 48, 2 and 24, etc., until you come to one that you have already used. For example, after matching 6 with 8, you move to 7 which is not a factor. The next number is 8 and you have already used it with 6, so you are finished. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions 21. 24 seconds PTS: OBJ: NAT: STA: TOP: KEY: ANS: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.1 Finding Common Multiples common multiples | least common multiple | common factors | greatest common factor 22. ANS: a. Carlos can make 32 sacks. b. Each sack would contain 3 small candy bars and 2 small popcorn balls. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.3 Finding Common Factors common multiples | least common multiple | common factors | greatest common factor 23. ANS: a. 6 b. Possible answers: 2 or 3 c. 210 d. Possible answers: 420, 630, 840 PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | factor tree | prime factorization 24. ANS: Odd; possible answers: is equivalent to or a string of odd factors. An odd number times an odd number is an odd number. This means that 32 is odd and 52 is odd, and when these two odd products are multiplied the final product will be odd. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Unit Test Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Finding the Longest Factor String factor tree | exponent | Fundamental Theorem of Arithmetic | odd numbers 25. ANS: Yes; the factors of numbers greater than 16 on the Factor Game board are between 1 and 15, so any number greater than 16 would be an illegal move because its factors are already circled. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors 26. ANS: Possible answer: It is an even number. It can be divided by 2 without a remainder. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors | multiple 27. ANS: Possible answer: It can be divided by 3 without a remainder. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors | multiple 28. ANS: Possible answer: It can be divided by 5 without a remainder. It ends in 0 or 5. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors | divisibility 29. ANS: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96; Possible answers: They can be divided by 6 without a remainder. They have 6 as a factor. They are divisible by 2 and 3. PTS: 1 DIF: L2 REF: Prime Time | Extra Questions OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: prime number | composite number | abundant numbers | perfect numbers | deficient numbers | common multiples | multiple 30. ANS: a. 225; to find all the factors of a number, you must check every whole number less than or equal to the square root of the number. If Mr. Matsumoto must check the numbers from 1 through 15, the number must be greater than or equal to , or 225, and less than , or 256. b. 255; as mentioned in the answer to part (a), the number must be less than , or 256. The greatest it could be is 255. PTS: OBJ: STA: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: prime number | composite number | abundant numbers | perfect numbers | deficient numbers | factor 31. ANS: 11 PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions 32. ANS: Answers will vary. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions 33. ANS: a. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 b. c. 4, 9, 25, 49 (the squares of primes) d. 121 PTS: OBJ: NAT: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns factor pair | rectangular model | dimensions | square numbers | factor 34. ANS: 81 PTS: OBJ: NAT: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns factor pair | rectangular model | dimensions | prime factorization | prime number 35. ANS: a. 1, 2, 10, 20; If a number has 4 and 5 as factors, it must have the factors of 4 and 5 as factors, namely 1, 2, 4, and 5. It must also have the products of 2 and 5 and of 4 and 5 as factors, since these pairs of factors do not have any common factors. b. The smallest number is 20, because 4 and 5 do not have any common factors. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor 36. ANS: a. 6 b. 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, and 96 PTS: OBJ: NAT: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: Problem 2.1 Finding Patterns factor pair | rectangular model | dimensions | factor | multiple 6PA 2.2.8.B 37. ANS: a. Ben bought four 9-ounce tubes. Aaron bought three 12-ounce tubes. b. The 12-ounce tube is the better buy at 9.6 cents per ounce. The 9-ounce tube cost .9 cents per ounce. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.1 Finding Common Multiples common multiples | least common multiple | common factors | greatest common factor 38. ANS: a PTS: 1 DIF: L2 REF: Prime Time | Extra Questions OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: divisible by | prime number | composite number | abundant numbers | perfect numbers | deficient numbers | divisibility 39. ANS: Both are correct but Tyrone's is the accepted form. When we make a factor string, we use only prime factors. Otherwise, the strings could go on forever. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.1 Finding Factor Strings strings of factors | factorization | prime factorization 40. ANS: 120; The first three prime numbers are 2, 3, and 5. The first three composite numbers are 4 = 2 multiply 2, 6 = 2 multiply 3, and 8 = 2 multiply 2 multiply 2. The shortest string that contains the factors of all these numbers is 2 multiply 2 multiply 2 multiply 3 multiply 5. The smallest number that is divisible by all the numbers is the product of this string, which is 120. PTS: 1 DIF: L2 REF: Prime Time | Extra Questions OBJ: Investigation 2: Whole-Number Patterns and Relationships NAT: NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B TOP: Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | prime number | divisible by | divisibility | composite number 41. ANS: The best move in this case would be 25, which gives your opponent only 5 points. Note: A prime number would be a bad move, since its only proper factor, 1, has already been circled. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors whole number factors | whole number divisors | factor | proper factors 42. ANS: Possible response: We played the Factor Game today. I discovered that it is best to go first and choose 29, the highest prime on the board, as your first move. After the first move, choose numbers like 25 that leave your opponent a small number of factors. Stay away from numbers like 30, which have many factors, until most of the factors are already circled. PTS: OBJ: STA: KEY: factors 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors composite number | prime number | whole number factors | whole number divisors | factor | proper 43. ANS: a. b. Possible answer: Two cookies in a bag would be affordable and is a number a student would typically eat. This would also allow more students to buy cookies. c. Possible answer: Each cookie cost $0.15 to make. They could be sold at $0.25 per cookie. So, a bag of one would cost $0.25, a bag of two would cost $0.50, … PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.4 Finding the Greatest Common Factor factor | greatest common factor | least common multiple | multiple 44. ANS: a. Alternate multiplying of terms by 2 and 3 to generate the next term. b. 1,512 and 4,536 c. 7, since it is the only prime number in the sequence and is a factor of all the other terms in the sequence no matter how many terms are added. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | greatest common factor | prime factorization | factor 45. ANS: a. Any two numbers that do not share a common factor (relatively prime numbers) will work. Examples are 3 and 4, 11 and 12, 15 and 8. b. Any two numbers that share a common factor will work. Examples are 15 and 9, 10 and 25, 18 and 48, 45 and 8l. c. If the numbers do not have a common factor, their least common multiple will be equal to their product. If the numbers have a common factor, their least common multiple will be less than their product. PTS: OBJ: NAT: STA: TOP: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding Least Common Multiple KEY: least common multiple | multiple 46. ANS: a. , or written as b. Answers will vary. Some possible answers: It is an even number. It is not an odd number. Nine is one of its factors. It is divisible by 15. It is not prime. It is composite because it has more than 2 factors. This is its one unique string of factorization (Fundamental Theorem of Arithmetic). It is a square number because you could group its prime factors to represent two of the same numbers multiplying themselves: multiplied by is the same as saying or 302, which is 900. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | Extra Questions Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Finding the Longest Factor String factor tree | exponent | Fundamental Theorem of Arithmetic 47. a. 65 ANS: b. 17 PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors prime number | whole number factors | whole number divisors | factor | proper factors 48. ANS: Yes; since 2 and 5 are factors of 10, any number that has 10 as a factor must also have 5 as a factor. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers factor | prime number | composite number | abundant numbers | perfect numbers | deficient numbers 49. ANS: No; for example, the number 35 has 7 as a factor, but since it is an odd number it does not have 2 as a factor. 2 and 7 are both prime factors of 14; for a number to be a factor of 2 and 7, it would also be a factor of 14. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers factor | prime number | composite number | abundant numbers | perfect numbers | deficient numbers 50. ANS: a. 29 and 30, which have a product of . b. These two numbers give the largest product because they are the largest numbers. c. 1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435 One way to find the factors is to test factors below 29 to identify factor pairs. We know that so the middle factor pair is . PTS: 1 DIF: L2 REF: Prime Time | AP Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: factor pair | factor | prime number | composite number | abundant numbers | perfect numbers | deficient numbers 51. ANS: Answers will vary, but the following are examples of correct answers: a. b. c. d. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.3 Finding Multiples factorization | factoring | factor | multiple | square numbers | divisible by | near-perfect numbers 52. a. 3, 5, 2 ANS: b. possible answer: 2, 8, 14 and 3, 12, 21 c. infinitely many PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.3 Finding Multiples multiple | square numbers | divisible by | near-perfect numbers 53. a. ; ; ; ANS: b. ; ; ; ; ; ; ; c. ; ; ; ; ; ; ; ; ; ; ; d. A number is called a factor when it is multiplied by another number to find a product. A number is called a divisor of a number when it divides the dividend evenly to find a quotient. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.3 Finding Multiples factor | multiple | square numbers | divisible by | near-perfect numbers 54. ANS: The dimensions are . The possible dimensions are , , , and . Only the rectangle has dimensions with a sum of 11. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor | factor pair | rectangular model | dimensions 55. ANS: The dimensions are . The possible dimensions are , , , , and . Only the rectangle has dimensions with a sum that is a prime number. PTS: OBJ: NAT: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns factor pair | rectangular model | dimensions | prime number | factor 56. ANS: a. , , , b. , , , c. d. , , , e. , , f. , , , , , g. , , , PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor 57. ANS: Phillip’s number is 18 since the factor pairs , , and have the required sums and 18 < 20. PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor 58. ANS: In order to determine the number of tiles in each of the rectangles, multiply the tiles along the length by the tiles on the width. a. 60 b. 40 c. 40 PTS: OBJ: NAT: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor 59. ANS: a. b. 1 is the only number in the intersection since 13 is a prime number. c. Answers will vary; examples include factors of two different prime numbers (e.g., 5 and 13) or factors of two different relatively prime numbers (e.g., 11 and 14). PTS: 1 DIF: L2 REF: Prime Time | AP Investigation 2 OBJ: Investigation 2: Whole-Number Patterns and Relationships NAT: NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B TOP: Problem 2.3 Classifying Numbers KEY: prime number | composite number | factor | Venn diagram | intersection | greatest common factor | least common multiple 60. ANS: a. b. The numbers in the intersection are the multiples of 10, which is 5 × 2. Every number that is a multiple of 10 must have 5 and 2 as factors since 5 × 2 = 10. c. 75 would be placed with the multiples of 5 since it is a multiple of 5 but not a multiple of 2. 90 would be placed in the intersection since it is a multiple of both 5 and 2. PTS: OBJ: NAT: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.3 Classifying Numbers multiple | Venn diagram | intersection | greatest common factor | least common multiple 61. ANS: The fourth number is even. Since the first three numbers were even, odd, and odd, respectively, the sum of these three will be even. Thus, an even number must be added to this even sum to produce an even number. PTS: OBJ: NAT: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 2 Investigation 2: Whole-Number Patterns and Relationships NAEP N5a| NAEP N5b| NAEP N5f STA: 6PA 2.2.8.B Problem 2.2 Reasoning with Even and Odd Numbers even numbers | odd numbers | conjecture 62. ANS: a. Susan shouldn’t have to wait at all. The #14 bus should arrive at the mall at 10 A.M. and the #11 bus should leave the mall for the museum at about 10 A.M. (since the #11 bus runs every 12 minutes, it leaves at the top of every hour). b. Susan will have to wait 12 minutes because the #14 bus should arrive at noon. c. Both buses are at the mall at 9 A.M., 10 A.M., 11 A.M., and noon because the least common multiple of 15 and 12 is 60. PTS: OBJ: NAT: STA: TOP: KEY: factor 1 DIF: L2 REF: Prime Time | AP Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.1 Finding Common Multiples factor | multiple | common multiples | least common multiple | common factors | greatest common 63. ANS: a. One rectangle is made with 42 tiles, and the other is made with 56 tiles. These are the only two even multiples of 7 between 40 and 60. b. The rectangle with 42 tiles has a length of 6, and the rectangle with 56 tiles has a length of 8. These answers are found by finding the other number in the factor pair with 7 for each rectangle. c. Students’ answers will vary. For 42: , , or . For 56: , , or . PTS: 1 DIF: L2 REF: Prime Time | AP Investigation 3 OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.1 Finding Common Multiples KEY: even numbers | factor | multiple | common multiples | least common multiple | common factors | greatest common factor 64. ANS: Jack will only have a conflict one day per month on the 21st. The least common multiple of 3 and 7 is 21. The next common multiple, 42, is greater than the number of days in a month. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding the Least Common Multiple factor | multiple | common multiples | least common multiple | common factors 65. ANS: a. Since the numbers are prime, they don’t have any proper factors other than 1. Therefore, their least common multiple would be their product. b. Since the numbers are prime, the only factors each number has is 1 and itself. Therefore, the greatest common factor must be 1. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding the Least Common Multiple | Problem 3.4 Finding the Greatest Common Factor common factors | least common multiple | multiple | prime number | greatest common factor 66. ANS: a. LCM: 24; GCF: 4 b. LCM: 105; GCF: 1 c. LCM: 187; GCF: 1 d. LCM: 108; GCF: 36 e. Parts (b) and (c), for part ((b), the two numbers are relatively prime. For part (c), the two numbers are prime. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding the Least Common Multiple | Problem 3.4 Finding the Greatest Common Factor factor | multiple | greatest common factor | least common multiple 67. ANS: a. 4 b. 5 c. 10 d. 25 e. The greatest common factor of two numbers is one of the two numbers when the smaller number is a factor of the larger number PTS: OBJ: NAT: STA: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.4 Finding the Greatest Common Factor KEY: factor | greatest common factor 68. ANS: a. b. c. d. e. 23 is prime. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.1 Finding Factor Strings factor | strings of factors | factorization | prime factorization 69. a. b. c. d. ANS: PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Finding the Longest Factor String factor tree | exponent | Fundamental Theorem of Arithmetic 70. ANS: a. GCF = 5, LCM = 525 b. GCF = 27, LCM = 81 c. GCF = 9, LCM = 252 PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | greatest common factor | least common multiple 71. ANS: 105; an odd number cannot have a factor of 2, and 3, 5, and 7 are the only three primes with a product less than 160. PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | odd numbers | prime number | prime factorization 72. ANS: 1,800 PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | prime factorization | exponent 73. ANS: a. Possible answer: 6 and 36 b. Possible answer 12 and 60 c. The least common multiple is the other number in the pair. PTS: 1 DIF: L2 REF: Prime Time | AP Investigation 4 OBJ: Investigation 4: Factorizations: Searching for Factor Strings NAT: NAEP N5b| NAEP N5d STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 4.3 Using Prime Factorizations KEY: prime factorization | abundant numbers | factor | relatively prime | greatest common factor | least common multiple 74. ANS: a. Yes, the only common factor they have is 1. b. Yes, the only common factor they have is 1. c. No, even numbers always have a factor of 2. d. Make sure that all the factors of the second number differ from the first (except for 1). PTS: OBJ: NAT: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | factor | common factors 75. ANS: Begin with the number 1, repeatedly multiply by 3 until the result exceeds 1,000. The numbers would be 3, 9, 27, 81, 243, 729. PTS: OBJ: STA: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors KEY: factor | prime factor 76. ANS: a. Alternate multiplying terms by 5 and then 3. In other words, multiply the first term by 5 to get the second term, multiply the second term by 3 to get the third, multiply the third term by 5 to get the fourth, multiply the fourth term by 3 to get the fifth, and so on. b. 3,375 and 10,125. c. The greatest common factor is 3, since it is the first term in the sequence and a prime number. PTS: OBJ: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors factor | multiple | greatest common factor 77. a. Student 1 ANS: b. Students 1, 2, and 4 PTS: OBJ: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors factor | multiple | greatest common factor | least common multiple 78. ANS: a. Locker 1,000 b. Locker 988 c. Locker 960 d. Locker 600 PTS: OBJ: STA: TOP: KEY: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors factor | multiple | greatest common factor | least common multiple 79. ANS: The largest prime desert less than 50 is {24, 25, 26, 27, 28}. PTS: OBJ: STA: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors KEY: factor | multiple | prime number 80. ANS: a. b. PTS: OBJ: STA: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors KEY: factor | multiple | prime number 81. a. b. c. ANS: d. PTS: OBJ: STA: TOP: 1 DIF: L2 REF: Prime Time | AP Investigation 5 Investigation 5: Putting It All Together NAT: NAEP N5b| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 5.1 Using Multiples and Factors KEY: factor | prime factorization 82. ANS: 71,094 is not divisible by 5. PTS: OBJ: STA: KEY: 1 DIF: L1 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers divisible | divisibility test | factor 83. ANS: 2, 3, 4 PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers divisible | divisibility test | composite number | prime number 84. ANS: PTS: OBJ: STA: TOP: KEY: 1 DIF: L1 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.1 Finding Factor Strings strings of factors | prime factorization | composite number | prime number | factor tree | factor a. 85. 2 b. 5 or 2 students in each group PTS: OBJ: STA: TOP: KEY: 1 DIF: L2 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.2 Using Prime Factorizations multi-part question | factor | factoring | prime number | word problem | problem solving ANS: MULTIPLE CHOICE OBJ: NAT: STA: TOP: 86. ANS: C PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.4 Finding the Greatest Common Factor KEY: factor | greatest common factor 87. ANS: B PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.3 Finding Common Factors KEY: factor OBJ: NAT: STA: TOP: 88. ANS: D PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.3 Finding Common Factors KEY: multiple OBJ: NAT: STA: TOP: KEY: 89. ANS: D PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.1 Finding Factor Strings strings of factors | factorization | prime factorization | factor 90. ANS: D PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.2 Finding the Least Common Multiple KEY: multiple | common multiples OBJ: NAT: STA: TOP: KEY: 91. ANS: D PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations relatively prime | prime factorization OBJ: NAT: STA: TOP: 92. ANS: C PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding the Least Common Multiple KEY: multiple | least common multiple 93. ANS: A PTS: 1 DIF: L2 REF: OBJ: Investigation 2: Whole-Number Patterns and Relationships NAT: NAEP N5a| NAEP N5b| NAEP N5f STA: TOP: Problem 2.2 Reasoning with Even and Odd Numbers KEY: even numbers | odd numbers | conjecture OBJ: NAT: STA: TOP: Prime Time | Multiple Choice 6PA 2.2.8.B 94. ANS: B PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations KEY: relatively prime | prime factorization | factor OBJ: NAT: STA: TOP: 95. ANS: C PTS: 1 DIF: L2 REF: Prime Time | Multiple Choice Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding the Least Common Multiple KEY: multiple | least common multiple 96. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisible | divisibility test | proper factors 97. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: divisible | divisibility test | factor | composite number | prime number 98. ANS: C PTS: 1 DIF: L1 REF: Skills Practice Investigation 3 OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.1 Finding Common Multiples | Problem 3.2 Finding the Least Common Multiple | Problem 3.3 Finding Common Factors | Problem 3.4 Finding the Greatest Common Factor KEY: multiple | least common multiple | prime factorization | factor 99. ANS: C PTS: 1 DIF: L1 REF: Skills Practice Investigation 3 OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5d STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 4.1 Finding Factor Strings | Problem 4.2 Finding the Longest Factor String | Problem 4.3 Using Prime Factorzations KEY: prime factorization | prime number | factoring | factor | factor tree | exponent 100. ANS: C PTS: 1 DIF: L1 REF: Skills Practice Investigation 4 OBJ: Investigation 4: Factorizations: Searching for Factor Strings NAT: NAEP N5b| NAEP N5d STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 4.1 Finding Factor Strings | Problem 4.2 Finding the Longest Factor String | Problem 4.3 Using Prime Factorizations KEY: prime factorization | prime number | factoring | factor | factor tree | exponent 101. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisibility | factor OBJ: STA: 102. ANS: B PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisibility | factor 103. ANS: A PTS: 1 DIF: L2 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisibility | factor 104. ANS: C PTS: 1 DIF: L2 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisible | divisibility test 105. ANS: D PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisible | divisibility test 106. ANS: D PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisible | divisibility test OBJ: NAT: STA: TOP: 107. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.2 Finding the Least Common Multiple KEY: multiple | least common multiple 108. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: composite number | prime number 109. ANS: B PTS: 1 DIF: L1 REF: Skills Practice Investigation 3 OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.4 Finding the Greatest Common Factor KEY: greatest common factor | prime factorization | factor OBJ: NAT: STA: TOP: KEY: 110. ANS: B PTS: 1 DIF: L2 REF: Skills Practice Investigation 4 Investigation 4: Factorizations: Searching for Factor Strings NAEP N5b| NAEP N5d 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 4.3 Using Prime Factorizations greatest common factor | prime factorization | factor | word problem | problem solving 111. ANS: D PTS: 1 DIF: L1 REF: OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f Skills Practice Investigation 3 STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.4 Finding the Greatest Common Factor KEY: greatest common factor | factor | factoring OBJ: NAT: STA: TOP: KEY: 112. ANS: C PTS: 1 DIF: L1 REF: Skills Practice Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.4 Finding the Greatest Common Factor division ladder | greatest common factor | factor | factoring 113. ANS: C PTS: 1 DIF: L2 REF: Skills Practice Investigation 3 OBJ: Investigation 3: Common Multiples and Common Factors NAT: NAEP N5b| NAEP N5c| NAEP N5f STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 TOP: Problem 3.4 Finding the Greatest Common Factor KEY: greatest common factor | factor | factoring OBJ: NAT: STA: TOP: KEY: 114. ANS: B PTS: 1 DIF: L1 REF: Skills Practice Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.4 Finding the Greatest Common Factor greatest common factor | factor | factoring OBJ: NAT: STA: TOP: KEY: 115. ANS: D PTS: 1 DIF: L3 REF: Skills Practice Investigation 3 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.3 Finding Common Factors greatest common factor | factor | factoring | word problem | problem solving OBJ: NAT: STA: TOP: 116. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 4 Investigation 3: Common Multiples and Common Factors NAEP N5b| NAEP N5c| NAEP N5f 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3 Problem 3.3 Finding Common Factors KEY: factor | factoring 117. ANS: A PTS: 1 DIF: L1 REF: Skills Practice Investigation 1 OBJ: Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.2 Prime and Composite Numbers KEY: composite number OTHER 118. ANS: No; the sum of the digits is not divisible by 9. PTS: OBJ: 1 DIF: L2 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d STA: 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors KEY: divisibility | writing in math 119. ANS: Since 1 + 0 + 8 = 9, and 9 is divisible by 9, 108 is divisible by 9. So Jennifer can divide her CDs evenly among her boxes. She can put 12 CDs in each box. Since 193 does not end in 0 or 5, 193 is not divisible by 5. So Melissa cannot divide her CDs evenly among her boxes. PTS: OBJ: STA: KEY: 1 DIF: L2 REF: Skills Practice Investigation 1 Investigation 1: Factors and Products NAT: NAEP N5b| NAEP N5c| NAEP N5d 6PA 2.2.8.B TOP: Problem 1.1 Finding Proper Factors divisible | divisibility test | word problem | problem solving | writing in math | reasoning