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mt2-f03-sol

mt2-f03-sol - CS 70 Fall 2003 P RINT your name Discrete...

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CS 70 Discrete Mathematics for CS Fall 2003 Wagner MT2 Soln PRINT your name: , (last) (first) SIGN your name: PRINT your username on cory.eecs : WRITE your section number (101 or 102): This exam is open-book, open-notes. No calculators are permitted. Do all your work on the pages of this examination. If you need more space, you may use the reverse side of the page, but try to use the reverse of the same page where the problem is stated. You have 80 minutes. There are 4 questions, worth from 20 to 30 points each (100 points total). The questions are of varying difficulty, so avoid spending too long on any one question. Do not turn this page until the instructor tells you to do so. Problem 1 Problem 2 Problem 3 Problem 4 Total CS 70, Fall 2003, MT2 Soln 1

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Problem 1. [True or false] (20 points) Circle TRUE or FALSE. You do not need to justify your answers on this problem. N denotes the set of natural numbers, { 0 , 1 , 2 ,... } . (a) TRUE or False : Let p be prime; then we’re guaranteed that x p - 1 1 ( mod p ) for all x N . [Counterexample: x 0 ( mod p ) .] (b) TRUE or False : Let p N be such that x p - 1 1 ( mod p ) holds for every x N with gcd ( x , p ) = 1; then p is guaranteed to be prime. [Counterexample: p = any Carmichael number.] (c) True or FALSE: Let p be prime and suppose x N satisfies x 6≡ 0 ( mod p ) ; then we’re guaranteed that x p 2 - p 1 ( mod p 2 ) . [Proof: ϕ ( p 2 ) = p 2 - p , since there are p multiples of p less than p 2 . If x 6≡ 0 ( mod p ) , then gcd ( x , p 2 ) = 1, and the result then follows from Euler’s theorem.] (d) TRUE or False : Let S , T be arbitrary sets; then we’re guaranteed that | S T | = | S | + | T | . [Counterexample: S = T = { 0 } .] (e) TRUE or False : Let A , B be events; then we’re guaranteed that Pr [ B | A ] = Pr [ A and B ] / Pr [ B ] . [Counterexample: any events where Pr [ A ] 6 = Pr [ B ] .] CS 70, Fall 2003, MT2 Soln 2
Problem 2. [Short answer] (30 points) Show your work on these problems. Circle your final answer. (a) You’ve been hired by the local phone company. They’re concerned, because all the local taxi companies have started demanding phone numbers made up of exactly 2 different digits. (For instance, “555-5556” and “811-1881” are acceptable, but “111-1111” and “123-4567” are not.) Your job is to help the phone

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• Fall '08