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Discrete Mathematics with Graph Theory (3rd Edition) 87

Discrete Mathematics with Graph Theory (3rd Edition) 87 -...

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Section 4.3 71"(500) = 95, 1;~gO ~ 80.456 and 50~)~~OJoo ~ 1.181. 13. There are approximately 5000/ In 5000 ~ 587 primes less than 5000, 50,000/ In 50,000 ~ 4621 primes less than 50,000, 500,000/In500,000 ~ 38,103 primes less than 500,000 and 5,000,000/ In 5,000,000 ~ 324,150 primes less than 5,000,000. 85 14. [BB] This is a special case of Exercise 11, Section 4.2 since if p and q are distinct primes, then p and q are relatively prime. The result also follows quickly from the Fundamental Theorem of Arithmetic since, writing n = P1P2 ... Pr, the hypotheses say that one of the Pi is P and some other Pi is q. 15. (a) Since 2xo + 5yo = 2x + 5y, we have 5(yo - y) = 2(x - xo). Since 5 divides the left side, 5 I 2(x - xo), so 5 I (x - xo) by Proposition 4.3.7. Thus x - xo = 5k (hence x = xo + 5k) for some integer k. From 5(yo - y) = 2(x - xo), we conclude that 5(yo - y) = 10k, so Yo - Y = 2k and y = Yo - 2k as desired. (b) This follows from part (a) since Xo = 2, Yo = 3 satisfies 2xo + 5yo = 19. 16. [BB] Without loss of generality, we may assume that a < b. Since there are infinitely many primes, there exists an integer n 2: 0 such that b + n is prime. Since a + n < b + n, b + n cannot divide a + n, so a + n and b + n are relatively prime.
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