9.3 Modular Arithmetic 567 57. Presidential Elections In the United States, presidential elections have been held every four years starting in 1788. Each of these years is congruent to 0 in modulo 4. a) List the first five presidential election years that occurred after 1788. 1792, 1796, 1800, 1804, 1808 b) List the first five presidential election years that will occur after 2020. 2024, 2028, 2032, 2036, 2040 c) List the presidential election years that will occur between 2550 and 2575. 2552, 2556, 2560, 2564, 2568, 2572 58. Governors’ Elections Wisconsin gubernatorial elections (elections for governor) have been held every four years starting in 1970. Each of these years is congruent to 2 in modulo 4. a) List the first five Wisconsin gubernatorial election years held after 1970. 1974, 1978, 1982, 1986, 1990 b) List the first five Wisconsin gubernatorial election years that will occur after 2020. 2022, 2026, 2030, 2034, 2038 c) List the five Wisconsin gubernatorial election years that will occur between 2100 and 2120. 2102, 2106, 2110, 2114, 2118 59. Golf Pro Schedule Jahlil is a golf pro who works at two different golf courses. His schedule is to work at Longboat Key golf course for 4 days, then work at The Meadows golf course for 3 days, and then rest for 2 days. He then repeats this cycle. If Jahlil is currently on his first day of rest, determine what he will be doing a) 30 days from today. Working at Longboat Key b) 100 days from today. Resting c) 366 days from today. Working at The Meadows 60. Flight Schedules A pilot is scheduled to fly for 5 consecutive days and rest for 3 consecutive days. If today is the second day of her rest shift, determine whether she will be flying or resting a) 60 days from today. Flying b) 90 days from today. Flying c) 240 days from today. Resting d) Was she flying 7 days ago? No e) Was she flying 20 days ago? Yes 7. 4 1+ 0 8. 2 6 + 3 9. 10 1− 4 10. 12 5 − 2 11. 8 9 ⋅ 2 12. 8 7 ⋅ 1 13. 4 8 − 1 14. 2 4 − 3 15. (15 4) 8 ⋅ − 2 16. (4 9) 7 − ⋅ 0 In Exercises 17–26, determine the modulo class to which each number belongs for the indicated modulo system. 17. 15, mod 2 1 18. 27, mod 4 3 19. 77, mod 8 5 20. 43, mod 6 1 21. 41, mod 9 5 22. 75, mod 8 3 23. 1, − mod 7 6 24. 7, − mod 4 1 25. 27, − mod 8 5 26. 39, − mod 7 3 In Exercises 27– 40, determine all positive number replacements (less than the modulus) for the question mark that make the statement true. 27. + ≡ ? 5 3 (mod 6) 4 28. + ≡ ? 3 2 (mod 5) 4 29. 6 ? 2 (mod 7) + ≡ 3 30. 4 ? 3 (mod 6) + ≡ 5 31. 4 ? 5 (mod 6) − ≡ 5 32. 3 ? 7 (mod 8) − ≡ 4 33. 4 ? 4 (mod 8) ⋅ ≡ 5 34. 4 ? 3 (mod 7) ⋅ ≡ 6 35. 2 ? 1 (mod 6) ⋅ ≡ {} 36. 3 ? 5 (mod 6) ⋅ ≡ {} 37. 6 ? 1 (mod 7) ⋅ ≡ 6 38. 3 ? 2 (mod 5) ⋅ ≡ 4 39. ? 6 3 (mod 7) − ≡ 2 40. − ≡ ? 3 4 (mod 8) 7 Problem Solving In Exercises 41– 48, assume that Sunday is represented as day 0, Monday is represented as day 1, and so on. If today is Thursday (day 4), determine the day of the week it will be in the specified number of days. Assume no leap years. 41. 17 days Sunday 42. 58 days Saturday 43. 105 days Thursday 44. 365 days Friday 45. 3 years Sunday 46. 4 years, 34 days Sunday 47. 400 days Friday 48. 5 years, 25 days Saturday In Exercises 49–56, consider the 12 months to be a modulo 12 system with January being month 0. If it is currently October, determine the month it will be in the specified number of months. 49. 22 months August 50. 49 months November 51. 2 years, 10 months August 52. 4 years, 8 months June 53. 83 months September 54. 7 years October 55. 105 months July 56. 5 years, 9 months July CarlosBarquero/Shutterstock
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