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hw708sol

# hw708sol - CS 1050 B Constructing Proofs Solutions to...

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CS 1050 B: Constructing Proofs April 12, 2008 Solutions to Homework 7 Lecturer: Sasha Boldyreva Problem 7.1, 12 points. Problem 18 from Section 6.2 of Rosen’s textbook. a) 1/7. b) Following the Birthday bound reasoning: Pr(at least 2 are born on the same day) = 1 - Pr(all born on different days) = 1 - 6 7 · 5 7 · . . . · 8 - n 7 . c) Plug in n= 2, 3 and so on, and see that the first time it exceeds 1/2 is when n=4, so this is the answer. You can use the Birthday bound too. Problem 7.2, 5 points. Problem 20 from Section 6.2 of Rosen’s textbook. The probability that none of n people has a birthday today is (366 / 366) n . We need to find the smallest n such that this is less than 1/2. We can find that the answer is 254 by trial-and-error or by using logarithms. Problem 7.3, 5 points. Problem 24 (d) and (e) from Section 3.6 of Rosen’s textbook. Show the intermediate results. d) gcd(1529,14039)=gcd(1529,278)=gcd(278,139)=gcd(139,0)=139. e) gcd(1529,14038)=gcd(1529,277)=gcd(277,144)=gcd(144,133)=gcd(133,11)=gcd(11,1)=gcd(1,0)=1. Problem 7.4, 5 points. Run the Extended GCD algorithm by hand on input (1529,14039) from the previous problem and show the intermediate and final results.

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hw708sol - CS 1050 B Constructing Proofs Solutions to...

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