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Evening Homework Problems Section 1 1. Calculate (314, 159) and find x and y such that 314x + 159y = 1. 2. Calculate (81, 24) and find x and y such that 81x + 24y = 3. 3. Is it true that for every positive integer a, (a, a + 1) = 1? If so, prove it. If not, find a specific counter-example. Section 2 1. Goldbach’s Conjecture (1742) states that every even integer greater than 2 can be expressed as the sum of 2 (not necessarily distinct) prime numbers. Express the numbers 48, 76 and 100 as the sum of two prime numbers. (This is an unsolved problem!) 2. A Mersenne Prime is a prime number of the form 2 n 1, where n 2 . Find the first 4 Mersenne primes. (On February 18, 2005, Dr. Martin Nowak from Germany found the new largest known prime number, 225,964,951 – 1. The prime number has 7,816,230 digits.) Section 3 1. Given the equation ax + by = d where (a, b) = d and x 0 , y 0 is an integer solution pair for this equation, verify that for all t Z, x x0 db t , y y 0 da t is an integer solution pair. Section 4 1. Find all m such that 1066 1776 (mod m). 2. Show that every prime (except 2) is congruent to 1 or 3 (mod 4). 3. A palindrome is a number that reads the same backward as forward. For example, 22, 1331, 27872. a. Prove that every four-digit palindrome is divisible by 11. b. Is every 5-digit palindrome divisible by 11? 4. Using the divisibility test for 7, show that 34489 is divisible by 7. 5. The International Standard Book Number (ISBN) is used to identify books. A correctly coded 10-digit ISBN a1a2 a3 a4 a5 a6 a7 a8 a9 a10 has the property that 10a1 9a2 8a3 7a4 6a5 5a6 4a7 3a8 2a9 a10 is divisible by 11. The incorrect ISBN 0-669-03925-4 is the result of a transposition of two adjacent Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1 digits not involving the first or last digit. Determine the correct ISBN. (Comment: all single errors and transposition errors can be detected for ISBNs.) Section 5 1. Solve 61x 7 (mod 126) by using the Extended Euclidean Algorithm, and list the smallest three positive solutions. 2. How many solutions are there? a. Is it possible for ax b (mod 20) to have exactly 7 solutions for some choice of a and b? b. Is it possible for ax b (mod 20) to have exactly 5 solutions for some choice of a and b? c. What possibilities are there for the number of solutions of ax b (mod 20)? Section 6 1. a. Calculate (n – 1)! (mod n) for n = 10, 12, 14 and 15. b. Based on your data in part a), make a conjecture. Speculate why it might be true. 2. Compute each of the following: a. 9119 (mod 4) b. the last digit of 7355 Section 7 1. Compute (77), (299) and (61). 2. Prove or disprove: If p is prime, then ( p 2 ) ( p 1) 2 . Many of the homework problems and the in-class problems come from several resources: 1. Joy of Numbers, by Judy Walker 2. Elementary Number Theory, 2nd Edition, by Underwood Dudley 3. For All Practical Purposes, 6th Edition, by COMAP 4. Cryptography notes from All Girls, All Math camp, by Judy Walker Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 2