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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.
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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.
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