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Discrete Mathematics with Graph Theory (3rd Edition) 80

# Discrete Mathematics with Graph Theory (3rd Edition) 80 -...

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78 (g) 8416 1 3719 0 978 1 785 -3 193 4 13 -19 11 270 2 -289 1 1715 (h) 100996 1 20048 0 756 1 392 -26 364 27 28 -53 (i) 28844 1 15712 0 13132 1 2580 -1 232 6 28 -67 8 542 4 -1693 G) 54321 1 12345 0 4941 1 2463 -2 15 5 3 -822 0 1 -2 7 -9 43 -611 654 -3881 0 1 -5 131 -136 267 0 1 -1 2 -11 123 -995 3108 0 1 -4 9 -22 3617 Solutions to Exercises gcd(8416, 3719) = 1 = 1715(8416) + (-3881) (3719), so gcd( -3719,8416) = 1 = 3881(-3719) + 1715(8416). So gcd(100996, 20048) = 28 = -53(100996) + 267(20048). gcd(28844,15712) = 4 = (-1693)(28844) + 3108(15712), so gcd(28844, -15712) = 4 = -1693(28844) + (-3108)( -15712). So, gcd(12345, 54321) = 3 = -822(54321) + 3617(12345). 13. The pairs (93,119), (-93,119), (-93, -119) and (3719,8416) are each relatively prime. 14. [BB] Since a and b are relatively prime, we have ma + nb = 1 for some integers m and n, so 2 = 2ma+2nb. Now suppose x 1 (a+b) and x 1 (a-b). Then, by Proposition 4.2.3, x 1 [(a+b)+(a-b)]; that is, x 12a. Also xl
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Unformatted text preview: [(a+b)-(a-b)]; that is, x 12b. Again, by Proposition 4.2.3, x 1 (2ma+2nb), so x 1 2 and we conclude that x = ±1 or x = ±2. The result follows. 15. [BB] If 91 and 92 are each greatest common divisors of a and b, then 91 ~ 92 (because 92 is greatest) and 92 ~ 91 (because 91 is greatest), so 91 = 92. 16. First note that a = b = 0 ¢:::::} a = a + b = 0, so gcd( a, b) exists if and only if gcd( a, a + b) exists. Suppose this is the case. As in the proof of Lemma 4.2.5, let 91 = gcd(a, a + b) and 92 = gcd(a, b). We prove that 91 ~ 92 and 92 ~ 91. First, since 92"1 a and 92 1 b, then 92 1 (a + b). ThUS,92 divides both a and a + b, so 92 ~ 91, the greatest common divisor of a and a + b. Also, since 91 divides both a and a + b, 91 must divide their difference, (a + b) -a = b. Thus, 91 divides both a and b, hence, 91 ~ 92, the greatest common divisor of a and b. 17. (a) [BB] Multiplying -2647(17369) + 8402(5472) = 1 by 4, we have 17369( -10588) + (5472)(33608) = 4....
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