Discrete Mathematics with Graph Theory (3rd Edition) 86

Discrete Mathematics with Graph Theory (3rd Edition) 86 -...

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84 Solutions to Exercises in the decomposition of ac as the product of primes are all even, that is, that ac = p~O:l p~O:2 ... p~O:r . It would follow that ac is the perfect square £2, where £ = p~l ... p~r. Suppose then that pO: is the largest power of a prime p dividing ac and that p2/3 is the largest power of p dividing b 2 , then the largest power of p dividing (nm)2 is po:+2f3. Since this power is even, a is even. It follows that ac is a perfect square, hence, a '" c. 8. (a) True. This is precisely Lemma 4.3.2. (b) [BB] False. The prime described here has the miraculous property that it divides all natural num- bers greater than 1. Certainly this prime cannot be 2 since 2 ~ 3, neither can it be odd since if p is an odd prime, p ~ 2. 9. (a) The range of 1 is the set of all prime numbers since any prime p is I( n) for some n: Take n = p, for example. (b) 1 is not one-to-one. For example, 1(2) = 2 = 1(4) but 2 #- 4. (c) Since the range is a proper subset of N,
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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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