222 CHAPTER 5 Number Theory and the Real Number System a) What size committees are possible? 4, 5, 10, 20, 25 b) How many committees are there for each size? 25, 20, 10, 5, and 4, respectively 57. Setting Up Chairs Jerrett is setting up chairs for his school band concert. He needs to put 120 chairs on the gymnasium floor in rows of equal size, and there must be at least 2 chairs in each row. List the number of rows and the number of chairs in each row that are possible. * 58. Barbie and Ken Mei collects Barbie dolls and Ken dolls. She has 390 Barbie dolls and 468 Ken dolls. Mei wishes to display the dolls in groups so that the same number of dolls are in each group and that each doll belongs to one group. If each group is to consist only of Barbie dolls or only of Ken dolls, what is the largest number of dolls Mei can have in each group? 78 dolls 59. Toy Car Collection Martha collects Matchbox and HotWheels toy cars. She has 70 red cars and 175 blue cars. She wants to line up her cars in groups so that each group has the same number of cars and each group contains only red cars or only blue cars. What is the largest number of cars she can have in a group? 35 cars 60. Stacking Trading Cards Desmond collects trading cards. He has 432 Magic: The Gathering cards and 360 Pokemon cards. He wants to make stacks of cards on a table so that each stack contains the same number of cards and each card belongs to one stack. If the Magic: The Gathering and Pokemon cards must not be mixed in the stacks, what is the largest number of cards that he can have in a stack? 72 cards 61. Tree Rows Elizabeth is the manager at Queen Palm Nursery and is in charge of displaying potted trees in rows. Elizabeth has 150 citrus trees and 180 palm trees. She wants to make rows of trees so that each row has the same number of trees and each tree is in a row. If the citrus trees and the palm trees must not be mixed in the rows, what is the largest number of trees that she can have in a row? 30 trees 62. Car Maintenance For many sport utility vehicles, it is recommended that the oil be changed every 3500 miles and that the tires be rotated every 6000 miles. If Carmella just had the oil changed and tires rotated on her SUV during the same visit to her mechanic, how many miles will she drive before she has the oil changed and tires rotated again during the same visit? 42,000 miles 63. Work Schedules Leah and Isaac both work the same night shift. Leah has every fifth night off and Isaac has every sixth Christian Mueller/Shutterstock night off. If they both have tonight off, how many days will pass before they have the same night off again? 30 days 64. Taking Medicine Elijah takes the medicine bisphosphonate once every 30 days and the medicine pegaspargase once every 14 days. If Elijah took both medicines on February 1, how many days would it be before he has to take both medicines on the same day again? 210 days 65. Relatively Prime Two numbers with a greatest common divisor of 1 are said to be relatively prime. For example, the numbers 9 and 14 are relatively prime, since their GCD is 1. Determine whether the following pairs of numbers are relatively prime. Write yes or no as your answer. a) 8, 9 Yes b) 15, 24 No c) 39, 52 No d) 177, 178 Yes 66. The primes 2 and 3 are consecutive natural numbers. Is there another pair of consecutive natural numbers both of which are prime? Explain. * 67. Determine the next two sets of twin primes that follow the set 11, 13. 17, 19, and 29, 31 68. Determine the first five Mersenne prime numbers. 3, 7, 31, 127, and 8191 (note: 2047 is not prime since 23 89 2047 ⋅ = ) 69. Determine the first three Fermat numbers and determine whether they are prime or composite. 5, 17, and 257 are all prime. 70. Show that Goldbach’s conjecture is true for the even numbers 4 through 20. * 71. State a procedure that defines a divisibility test for 14. A number is divisible by 14 if both 2 and 7 divide the number. 72. State a procedure that defines a divisibility test for 22. A number is divisible by 22 if both 2 and 11 divide the number. Euclidean Algorithm Another method that can be used to determine the greatest common divisor is known as the Euclidean algorithm. We illustrate this procedure by determining the GCD of 60 and 220. First divide 220 by 60 as shown below. Disregard the quotient 3 and then divide 60 by the remainder 40. Continue this process of dividing the divisors by the remainders until you obtain a remainder of 0. The divisor in the last division, in which the remainder is 0, is the GCD. ) ) ) 20 0 60 220 180 40 3 40 60 40 20 1 40 40 2 Since 40 20 / had a remainder of 0, the GCD is 20. In Exercises 73–78, use the Euclidean algorithm to determine the GCD. 73. 18, 24 6 74. 20, 35 5 75. 35, 105 35 76. 78, 104 26 77. 150, 180 30 78. 210, 560 70 *See Instructor Answer Appendix
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