Discrete Mathematics with Graph Theory (3rd Edition) 108

Discrete Mathematics with Graph Theory (3rd Edition) 108 -...

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106 Solutions to Review Exercises (c) I {x E Z I X 2 - 12 is divisible by 3} {x E Z I x 2 - 1 = 3k for some integer k} {x E Z I x-I = 3Uor some t' E Z or x + 1 = 3Uor some t' E Z} (3Z + 1) U (3Z - 1) (Since x 2 - 1 = (x + l)(x - 1), if 31 (x 2 - 1), then 3 I (x - 1) or 3 I (x + 1).) 16. (a) Suppose there are just a finite number of primes, PI, P2, ... , Pn. Then look at the number PIP2 ... Pn + 1. This can't be prime since it is bigger than each Pi. So it is divisible by some prime, hence by some Pi. But this implies Pi 11, which is not true. (b) Every integer n 2: 2 can be written in the form n = PIP2 ... Pr for unique primes PI. P2,···, Pr; equivalently, every integer n 2: 2 can be written n = qfl q~2 ... q':B as the product of powers of distinct prime numbers ql, q2, ... , qs. These primes and the exponents aI, a2, ... , as are unique. 17. 2 119 -1 = 2(17)(7) -1 = (217 _1)(2 102 + 2 85 + 2 68 + 2 51 + 2 34 + 217 + 1) is not prime; 3 109 -1 is not prime because it is even; 4 109 -1
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