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Discrete Mathematics with Graph Theory (3rd Edition) 93

Discrete Mathematics with Graph Theory (3rd Edition) 93 -...

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Section 4.4 91 Exercises 4.4 1. (a) [BB] 5,12,19, -2, -9, -16 are in 5. 4,11,18, -3, -10, -17 are in -3. (b) [BB] The general element of 5 is an integer of the form 7k + 5 for some integer k. The general form of an element in -3 is an integer of the form 7k - 3 for some integer k. 2. (a) 3,16,29,42, -10, -23, -36, -49, -62 are elements of 3. 11,24,37,50, -2, -15, -28, -41, -54 are elements of -2. (b) The general integer in 3 is of the form 13k + 3 for some integer k. The general integer in -2 is of the form 13k - 2 for some integer k. 3. (a) [BB] 1286 = 32(39) + 38, so 1286 (mod 39) = 38. (b) 43,197 = 129(333) + 240, so 43,197 (mod 333) = 240. (c) [BB] -545,608 = -10,699(51) + 41, so -545,608 (mod 51) = 41. (d) -125,617 = -399(315) + 68, so -125,617 (mod 315) = 68. (e) 11,111,111,111 = 10,001,000(1111) + 111, so 11,111,111,111 (mod 1111) = 111. 4. (a) [BB] False: 18 - 2 = 16 is not divisible by 10. (c) [BB] True: 44 - (-8) = 52 is divisible by 13. (b) True: -13 - 7 = -20 is divisible by 5. (d) False: 423 - 17 = 406 is divisible by 29, so 17 = 423. (e) False: 400 - (-18) = 418 is divisible by
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