A divisibility rule is a heuristic for determining whether a positive integer can be evenly divided by another i. If you dont know the new number s divisibility, you can apply the rule again. Introduction to number theory lecture notes university of san. Divisibility getting started many commands in this example worksheet are available at maples top level, meaning that no packages are.
One of the unique characteristics of these notes is the careful choice of topics and its importance in the theory of numbers. The freedom is given in the last two chapters because of the advanced nature of the topics that are presented. For example, determining if a number is even is as simple as checking to see if its last digit is 2, 4, 6, 8 or 0. Divisibility by 7 when the units digit is doubled and subtracted from the number formed by the remaining digits, the resulting number must be divisible by 7. Divisibility is the property of an integer number to be divided by another, resulting an integer number where a and b, two integers numbers, we will say that a is a multiple of b if there is an integer c, such as, when we multiply it by b is equal to a. Divisibility rules and examples showing how to use the rules. If a divides b, we also say a is a factor of b or b is a multiple of a and we write a. Starting with the essentials, the text covers divisibility, powers of integers.
Divisibility when dividing an integer by a second nonzero integer, the quotient may or may not be an integer. Andrews, evan pugh professor of mathematics at pennsylvania state university, author of the wellestablished text number theory first published by saunders in 1971 and reprinted by dover in 1994, has led an active career discovering fascinating phenomena in his chosen field number theory. To find out, if a number is divisible by, take the last digit, multiply it with 4, and add it to the rest of the number. It covers the basic background material that an imo student should be familiar with. The book uses the following notation for the quotient and remainder. Put the marbles in 2 piles with no leftovers put the marbles in 5 piles with no leftovers. The main audience will consist of olympiadlevel students. Find multiples for a given number divisibility tests to use sets of numbers to find and describe number patterns problems that can be solved with number theory. This number is not divisible by any of the pi it has remainder 1. Number theory is used to find some of the important divisibility tests, whether a given integer m divides the integer n. The true nature of number theory emerges from the first definition. This book was written as a text for the learning of number theory, not as a referente. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the chinese remainder theorem, diophantine equations, binomial coefficients. Divisibility in this book, all numbers are integers, unless speci.
Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic. Browse other questions tagged elementary number theory or ask your own question. I recommend this friendly volume for students looking for challenging problems in number theory and teachers of number theory for undergraduates. Divisibility rules from 1 to division rules in maths. In the remainder of the book we will concern ourselves principally with integers, and. For the first test, note that mod 7, 11, 103, 1002, 6, etc. The pace is fast, but the exposition in each chapter is detailed. Accordingly the word number, loosely used or symbolically denoted throughout this book, will be understood an integer, unless otherwise stated. The book could be used as a text for undergraduates.
Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. We try to prove it from first principles, that is, using only the definition of divisibility. God made the integers, all else is the work of man. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics. We say that a divides b if there is an integer k such that ak b. Thus in the next definition, d, n, and k are integers. Number theory is not like that, and for many problems, some of them ancient, we have more conjectures than theorems. Download for offline reading, highlight, bookmark or take notes while you read number theory. More formally, for a 6 0 we say that divides b if there is. Elementary number theory divisibility mathematics stack.
Divisibility and the division algorithm mathematics. Examples of proving divisibility statements by mathematical induction. This introductory textbook takes a problemsolving approach to number theory, situating each concept within the framework of an example or a problem for solving. Here are a few examples of number theoretic problems that have been solved. This course starts at the very beginning covering all of the essential tools and concepts in number theory, and then applying them to computational art, cryptography codebreaking, challenging logic puzzles, understanding infinity, and more. I structures, examples, and problems 1 divisibility 15 1. Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integervalued functions. If a and b are integers and there is some integer c such that a b c, then we say that b divides a or is a factor or divisor of a and write ba. Every time you buy a book from amazon, check your grades. We say that a divides b write a jb i there exists c 2z such that b ac.
Search for wildcards or unknown words put a in your word or phrase where you want to leave a placeholder. The book covers the usual topics of introductory number theory. Number theory have countless applications in mathematics as well in practical applications such as. For swedish and norwegian students i recommend reading and using the following book for more in depth theory, examples and exercises. The last digit is 0, so the number is divisible by 2. In this lesson, we are going to prove divisibility statements using mathematical induction. The exposition moves systematically and intuitively to uncover deeper properties. Divisibility, the fundamental theorem of number theory. Give an example of an infinite increasing arithmetic pro. The reason is students who are new to the topic usually start with problems involving summations followed by. Structures, examples, and problems ebook written by titu andreescu, dorin andrica. Number theory is a newly translated and revised edition of the most popular introductory textbook on the subject in hungary. Multiple divisibility rules applied to the same number in this way can help quickly determine its prime factorization without. Discrete mathematics introduction to number theory 235 divisibility i given two integers a and b where a 6 0, we say a divides b if.
The notation is read a divides b, which is a statement. Search within a range of numbers put between two numbers. Here are some of the most important number theory applications. If you get an answer divisible by including zero, then the original number is divisible by. If one number divides a second and the second number divides a third, then the first number divides the third. Here are a few examples of numbertheoretic problems that have been solved only recently or still remain open. The notation means that a divides b for example, since. Kens book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. In this book, all numbers are integers, unless specified otherwise. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics.
Divisibility millersville university of pennsylvania. The test of divisibility for 12 is that the number should be divisible by both 3 and 4. Mehdi hassani, the mathematical association of america, june, 2009. Divisibility theory mathematical exercises bioprofe. Division given two integers, say a and b, the quotient ba may or may not be an integer e. Divisibility trans4mind personal development resources.
Kens book is packed with examples and explanations that enable. By the word integers we mean the counting numbers 1, 2, 3. If a and b are integers and there is some integer c such that a bc, then we say that b divides a or is a factor or divisor of a and write ba. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. A number is divisible by 3 if the sum of its digits is divisible by 3. I number theory has a number of applications in computer science, esp. Find rules for divisibility for numbers where the divisor is 2, 4, 8, 5, or 25. Divisibility two useful properties of divisibility are one of the most useful properties of divisibility is that it is transitive. The last three digits are 000 which is divisible by 125, so the number is divisible by 53.
Divisibility by 6 it must be even and divisible by 3. The book covers a broad range of important topics that start from basic divisibility and congruences and move to quadratic residues, padic valuations and systems of congruences. Number theoryelementary divisibility wikibooks, open books for. Number theory explore the powers of divisibility, modular arithmetic, and infinity. Perhaps remind students of the divisibility test for 9 before presenting these.
Several examples are given and there is an indepth discussion. Allow for removal by moderators, and thoughts about future. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the chinese remainder theorem, diophantine equations, binomial coefficients, fermat and mersenne primes and other special numbers, and special sequences. Number theory structures, examples, and problems titu. The book is a collection of number theory problems chosen from various national and international mathematical olympiads. Basic concepts in number theory and its application in crypto instructor. If you are teaching a number theory course, you may wish to assign their proofs as an exercise. For example, the numbers 20, 36, and 45 are relatively prime, yet any two. The following theorems illustrate a number of important properties of divisibility. These books are both excellent sources of examples, additional practice. In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to. Divisibility number theory is concerned with the properties of the integers. Use the definition of divisibility to show that given any integers a, b, and c, where a. Here are a few examples of number theoretic problems that have been solved only recently or still remain open.
A number is divisible by 2 if its last digit is even or the last digit is 0,2,4,6,or 8. A rigorous exposition starts with the natural numbers and the basics. Number theorists study prime numbers as well as the properties of. Number theory divisibility tests number theory contents see also tables of.
For instance, 8596742 is divisible by 2 because the last digit is 2. A representation of an integer as a product of powers of distinct primes, i. If a and b are integers, a divides b if there is an integer c such that. For example, the numbers 20, 36, and 45 are relatively prime. We say that a divides b, or a is a divisor of b, or b is a multiple of a, if there exists an integer m such that b am. What is the least number of marbles that can satisfy the following situation. Nevertheless, new methods and results emerge in ts and starts, and the subject has seen many great advances in just the last few decades.
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