Discrete Mathematics with Graph Theory (3rd Edition) 102

Discrete Mathematics with Graph Theory (3rd Edition) 102 -...

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100 Solutions to Exercises (c) Suppose the digits at and aj of al ... ak are transposed. (Without loss of generality, assume i < j.) Let a = (al,"" ai, . .. , aj,'" ak) and a' = aj,"" ai,'" ak). Ifw· a' = w· a, then Wiaj + Wjai = Wiai + wjaj, so (Wi - wj)(aj - ai) == 0 (mod n). The assumption tells us that == ai n) and, since n > 9, we conclude that = ai. 18. (a) [BB] 1 = (-4)5+3(7), so x = 4(-4)(5) +3(3)(7) = -17 == 18 (mod 35). Therefore, x = 18. (b) 1 = (-2)4 + 1(9), so x = 8(-2)(4) + 1(1)(9) = -55 == 17 (mod 36). Therefore, x = 17. (c) 1 = (-3)5 + 2(8), so x = 7( -3)(5) + 3(2)(8) = -57 == 23 (mod 40). Therefore, x = 23. (d) [BB] 1 = (-3)8 + 1(25), so x = 17( -3)(8) + 6(1)(25) = -258 == 142 (mod 200). Therefore, x = 142. (e) 1 = 48(1917) - 239(385), so x = 75(48)(1917) - 3(239)(385) = 6,625,155 == 720,795 (mod 738,045). Therefore, x = 720,795. (f) 1 = -2647(17369) + 8402(5472), so x = 2974(-2647)(17369)"+ 1003(8402)(5472) = 46,113,671,232 - 136,731,859,682 = -90,618,188,450 Then, since -90,618,188,450 =
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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