Discrete Mathematics with Graph Theory (3rd Edition) 94

Discrete Mathematics with Graph Theory (3rd Edition) 94 -...

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92 Solutions to Exercises 8. [BB] Observe that 5° == 1 (mod 7), 51 == 5 (mod 7), 52 == 4 (mod 7), 53 == 6 (mod 7), 54 == 2 (mod 7) and 55 == 3 (mod 7). Thus, each of the integers 1,2,3,4,5,6 is congruent mod 7 to some power of 5. By Proposition 4.4.5, any integer a is congruent mod 7 to one of 0,1,2,3,4,5,6. So the result follows. 9. (a) [BB] No x exists. The values of 3x mod 6 are 0 and 3. (b) x = 2, x = 5; (c) x = 6. (d) No x exists. The values of 4x mod 6 are 0, 4 and 2. (e) [BB] If 50 1 (2x -18), then 2x -18 = 50k, so x = 9 + 25k for some integer k. We obtain x = 9, x = 34. (f) [BB] By trial and error, x = 9, or using the method of Problem 25, write 11-2(5) = 1, deducing that -2(5) == 1 (mod 11) and so x = -2(1) == 9 (mod 11). (g) If 25 1 (5x - 5), then 5x - 5 = 25k so x = 1 + 5k for some integer k. So x = 1,6,11,16,21. (h) No x exists. If 5921 (4x - 301), then 4x - 301 is even. This is impossible since, for any x, 4x is even. (i) No
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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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