202 CHAPTER 4 Systems of Numeration 23. 1101 111 2 2 − 1102 24. − 1001 110 2 2 112 25. 4223 304 7 7 − 36167 26. 2173 1654 8 8 − 3178 27. 4232 2341 5 5 − 13415 28. 2100 1012 3 3 − 10113 31. 12 2 3 3 × 1013 32. × 23 3 4 4 2014 29. A3B3 21B4 12 12 − 81BB12 30. 4E7 189 16 16 − 35E16 33. 647 5 8 8 × 41038 34. 124 6 12 12 × 72012 In Exercises 31– 42, multiply in the indicated base. 41. 316 16 7 7 × 60317 42. 7A9 12 12 12 × 926612 In Exercises 43–54, divide in the indicated base. 43. ) 2 312 4 4 1234 44. ) 2 730 8 8 3548 45. ) 3 506 7 7 146 R2 7 7 46. ) 4 302 5 5 34 R1 5 5 47. ) 3 200 8 8 52 R2 8 8 48. ) 4 252 8 8 52 R2 8 8 49. ) 2 213 4 4 103 R1 4 4 50. ) 5 214 6 6 24 R2 6 6 51. ) 3 224 5 5 41 R1 5 5 52. ) 4 210 6 6 31 R2 6 6 53. ) 6 404 7 7 45 R2 7 7 54. ) 3 2101 7 7 500 R1 7 7 35. 37 21 8 8 × 10178 36. 512 23 6 6 × 210206 37. B12 83 12 12 × 7767612 38. 6A3 24 12 12 × 13BB012 39. 110 11 2 2 × 100102 40. 111 101 2 2 × 1000112 Problem Solving In Exercises 55–58, suppose the numerals in a base 5 numeration system are as illustrated with their equivalent Hindu–Arabic numerals. 5 0 5 1 5 2 5 3 5 4 Add the following base 5 numerals. 55. 1 5 5 * 56. 1 5 5 * 57. 1 5 5 * 58. 1 5 5 * In Exercises 59– 66, assume the numerals given are in a base 4 numeration system. In this system, suppose colors are used as numerals, as indicated below. = 0, = 1, = 2, = 3 Add the following base 4 numerals. Your answers will contain a variety of the colors indicated. 59. 4 4 1 * 60. 1 4 4 * 61. 1 4 4 * 62. 1 4 4 * Subtract the following in base 4. Your answer will contain a variety of the colors indicated. 63. 2 4 4 * 64. 2 4 4 * 65. 2 4 4 * 66. 2 4 4 * Challenge Problems/Group Activities In Exercises 67– 68, perform the indicated operation. 67. × FAB 4 16 16 3EAC16 68. ) D FACE 16 16 134A RC 16 16 69. Consider the multiplication × 462 35 8 8 a) Multiply the numerals in base 8. 212528 b) Convert 4628 and 358 to base 10. 306 and 29 c) Multiply the base 10 numerals determined in part (b). 8874 d) Convert the answer obtained in base 8 in part (a) to base 10. 8874 e) Are the answers obtained in parts (c) and (d) the same? Why or why not? Yes 70. Determine b, by trial and error, if = 1304 204. b 5 71. In a base 4 system, suppose each of the four numerals is represented by one of the following colors: * Determine the value of each color if the following addition is true in base 4. 1 4 4 4 *See Instructor Answer Appendix
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