568 CHAPTER 9 Mathematical Systems c) Is there an identity element in the set? If so, what is it? Yes, 0 d) Does every element in the set have an inverse? If so, list the elements and their inverses. Yes; 0–0, 1–2, 2–1 e) The associative property holds for the system. Give an example. (12)2 1(22) + + = + + f) Does the commutative property hold for the system? Give an example. Yes; 2 1 1 2 + = + g) Is the system a commutative group? Yes h) Will every modulo system under the operation of addition be a commutative group? Explain. Yes 66. Construct a modulo 4 addition table. Repeat parts (b)–(h) in Exercise 65. * 67. a) Construct a modulo 4 multiplication table. * b) Is the system closed under the operation of multiplication? Yes c) Is there an identity element in the set? If so, what is it? Yes, 1 d) Does every element in the set have an inverse? Make a list showing the elements that have a multiplicative inverse and list the inverses. No; no inverse for 0 or for 2, inverse of 1 is 1, inverse of 3 is 3 e) The associative property holds for the system. Give an example. (12)3 1(23) × × = × × f) Does the commutative property hold for the system? Give an example. Yes, 2 3 3 2 × = × g) Is this mathematical system a commutative group? Explain. No, 0 and 2 do not have an inverse. 68. Construct a modulo 7 multiplication table. Repeat parts (b)–(g) in Exercise 67. * Challenge Problems/Group Activities We have not discussed division in modular arithmetic. With what number or numbers, if any, can you replace the question marks to make the statement true? The question mark must be a positive number less than the modulus. (Hint: Use the fact that a b c mod m ( ) ≡ means a b c mod m ( ) ≡ ⋅ and use trial and error to obtain your answer.) 69. ÷ ≡ ? 3 4 (mod 5) 2 70. 3 4 ? ÷ ≡ (mod 7) 6 71. ÷ ≡ ? ? 1 (mod 4) 1, 2, 3 72. 1 2 ? (mod 5) ÷ ≡ 3 In Exercises 73–75, solve for x where k is any counting number. 73. k x 8 ≡ (mod 8) 0 74. k x 7 1+ ≡ (mod 7) 1 75. k x 4 2 − ≡ (mod 4) 2 76. Determine the smallest positive number divisible by 5 to which 2 is congruent in modulo 6. 20 61. The Weekend Off A manager of a retail store has both Saturday and Sunday off every 7 weeks. It is week 2 of the 7 weeks. a) Determine the number of weeks before the manager will have both Saturday and Sunday off. 5 b) Will the manager have both Saturday and Sunday off 25 weeks from this week? No c) What is the first week, after 50 weeks from this week, that the manager will have both Saturday and Sunday off? 54 weeks from this week 62. Physical Therapy A patient has an Achilles’ tendon injury and is receiving physical therapy. The patient must have physical therapy twice a day for 4 days, physical therapy once a day for 2 days, and then 2 days of rest; then the cycle begins again. If the patient is in his second day of his twicea-day therapy cycle, determine what the patient will be doing a) 20 days from today. Once a day therapy b) 46 days from today. Resting c) 107 days from today. Once a day therapy d) Will the patient be resting 78 days from today? Yes 63. Nursing Shifts A nurse’s work pattern at Community Hospital consists of working the 7 a.m. 3− p.m. shift for 3 weeks and then the 3 p.m. –11 p.m. shift for 2 weeks. a) If it is the third week of the pattern, what shift will the nurse be working 6 weeks from now? 3 p.m. –11 p.m. b) If it is the fourth week of the pattern, what shift will the nurse be working 7 weeks from now? 7 a.m. –3 p.m. c) If it is the first week of the pattern, what shift will the nurse be working 11 weeks from now? 7 a.m. –3 p.m. 64. Driving Rotation A truck driver works both daytime and evening shifts. He works daytime for 5 consecutive days, then evenings for 3 consecutive days, then daytime for 4 consecutive days, then evenings for 2 consecutive days. Then the rotation starts again. If it is day 2 of the 5-day consecutive daytime shift, determine whether he will be working the daytime or evening shift a) 20 days from today. Evening b) 52 days from today. Day c) 365 days from today. Day 65. a) Construct a modulo 3 addition table. * b) Is the system closed? Explain. Yes *See Instructor Answer Appendix Cathy Yeulet/123RF
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