CSci 5471: Modern Cryptography, Spring 2010 Homework #1Yongdae Kim–Due: Feb. 2 9:00 AM–Type up your solution (either text or pdf) and send it to me and TA by email. (kyd(atmark)cs.umn.edu,yongdaek(atmark)gmail.com,hkang(atmark)cs.umn.edu)–Show all steps. Please ask any questions, if you find any problems.–Total 100 points + 20 points extra credit1.Discrete Math and Number Theory(a) (5 pts) Prove or disprove:n3+ (n+ 1)3+ (n+ 2)3is always mutiple of9. (Hint) Proof by cases(b) (5 pts) Show that if2n-1is a prime, thennis a prime. (Hint) Proof by contradiction.(c) (5 pts) Show that for all integersn,46 |n2+ 2.(d) (10 points) Letnbe a positive integer. The Euler functionφ(n)is defined as the number of non-negativeintegerskless thannwhich are relatively prime ton. Prove thati. (3 points)φ(p) =p-1for all primep.ii. (7 points)φ(pc) =pc-1(p-1)for all primepand all positive integerc.(e) (10 pts) Show that in any set ofn+ 1positive integers not exceeding2n, there must be two integers that arerelatively prime. (Hint) Use Pigeonhole Principles.
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