104 Number Theory Problems From the Training of the USA IMO Team
This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and research in number theory. Offering inspiration and intellectu
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umber Theory Problems From the Training of the USA IMO Team
Birkh¨auser Boston • Basel • Berlin
Titu Andreescu The University of Texas at Dallas Department of Science/Mathematics Education Richardson, TX 75083 U.S.A. [email protected]
Dorin Andrica “Babes¸-Bolyai” University Faculty of Mathematics 3400 Cluj-Napoca Romania [email protected]
Zuming Feng Phillips Exeter Academy Department of Mathematics Exeter, NH 03833 U.S.A. [email protected]
Cover design by Mary Burgess. Mathematics Subject Classification (2000): 00A05, 00A07, 11-00, 11-XX, 11Axx, 11Bxx, 11D04 Library of Congress Control Number: 2006935812 ISBN-10: 0-8176-4527-6 ISBN-13: 978-0-8176-4527-4
e-ISBN-10: 0-8176-4561-6 e-ISBN-13: 978-0-8176-4561-8
Printed on acid-free paper. c 2007 Birkh¨auser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
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104 Number Theory Problems Titu Andreescu, Dorin Andrica, Zuming Feng October 25, 2006
Contents Preface
vii
Acknowledgments
ix
Abbreviations and Notation
xi
1
1 1 4 5 7 11 12 13 16 17 18 19 24 27 33 36 38 40 46 52 65 70 71 72
Foundations of Number Theory Divisibility Division Algorithm Primes The Fundamental Theorem of Arithmetic G.C.D. Euclidean Algorithm B´ezout’s Identity L.C.M. The Number of Divisors The Sum of Divisors Modular Arithmetics Residue Classes Fermat’s Little Theorem and Euler’s Theorem Euler’s Totient Function Multiplicative Function Linear Diophantine Equations Numerical Systems Divisibility Criteria in the Decimal System Floor Function Legendre’s Function Fermat Numbers Mersenne Numbers Perfect Numbers
vi
Contents
2
Introductory Problems
75
3
Advanced Problems
83
4
Solutions to Introductory Problems
91
5
Solutions to Advanced Problems
131
Glossary
189
Further Reading
197
Index
203
Preface This book contains 104 of the best problems used in the training and testing of the U.S. International Mathematical Olympiad (IMO) team. It is not a collection of very difficult, and impenetrable questions. Rather, the book gradually builds students’ number-theoretic skills and techniques. The first chapter provides a comprehensive introduction to number theory and its mathematical structures. This chapter can serve as a textbook for a short course in number theory. This work aims to broaden students’ view of mathematics an
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