For the first test, note that mod 7, 11, 103, 1002, 6, etc. For instance, 8596742 is divisible by 2 because the last digit is 2. It covers the basic background material that an imo student should be familiar with. Examples of proving divisibility statements by mathematical induction. The exposition moves systematically and intuitively to uncover deeper properties.
If you are teaching a number theory course, you may wish to assign their proofs as an exercise. Here are a few examples of numbertheoretic problems that have been solved only recently or still remain open. Divisibility getting started many commands in this example worksheet are available at maples top level, meaning that no packages are. The main audience will consist of olympiadlevel students. The last digit is 0, so the number is divisible by 2. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. Every time you buy a book from amazon, check your grades.
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. 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. Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. 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. This number is not divisible by any of the pi it has remainder 1. Give an example of an infinite increasing arithmetic pro. Mehdi hassani, the mathematical association of america, june, 2009. We try to prove it from first principles, that is, using only the definition of divisibility. 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.
Perhaps remind students of the divisibility test for 9 before presenting these. In this lesson, we are going to prove divisibility statements using mathematical induction. Divisibility rules from 1 to division rules in maths. The book uses the following notation for the quotient and remainder. The test of divisibility for 12 is that the number should be divisible by both 3 and 4.
Divisibility rules and examples showing how to use the rules. Divisibility trans4mind personal development resources. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics. 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 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. The book could be used as a text for undergraduates. The last three digits are 000 which is divisible by 125, so the number is divisible by 53. Here are some of the most important number theory applications. Browse other questions tagged elementary number theory or ask your own question. Divisibility when dividing an integer by a second nonzero integer, the quotient may or may not be an integer. Allow for removal by moderators, and thoughts about future. 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. What is the least number of marbles that can satisfy the following situation.
Put the marbles in 2 piles with no leftovers put the marbles in 5 piles with no leftovers. A representation of an integer as a product of powers of distinct primes, i. For swedish and norwegian students i recommend reading and using the following book for more in depth theory, examples and exercises. Divisibility in this book, all numbers are integers, unless speci. Divisibility millersville university of pennsylvania. More formally, for a 6 0 we say that divides b if there is. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. This introductory textbook takes a problemsolving approach to number theory, situating each concept within the framework of an example or a problem for solving. 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. One of the unique characteristics of these notes is the careful choice of topics and its importance in the theory of numbers. 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 theory have countless applications in mathematics as well in practical applications such as.
The book is a collection of number theory problems chosen from various national and international mathematical olympiads. Number theory is used to find some of the important divisibility tests, whether a given integer m divides the integer n. If you get an answer divisible by including zero, then the original number is divisible by. In the remainder of the book we will concern ourselves principally with integers, and. By the word integers we mean the counting numbers 1, 2, 3. Here are a few examples of number theoretic problems that have been solved. The following theorems illustrate a number of important properties of divisibility. Number theoryelementary divisibility wikibooks, open books for. If a divides b, we also say a is a factor of b or b is a multiple of a and we write a.
Number theory is a newly translated and revised edition of the most popular introductory textbook on the subject in hungary. I number theory has a number of applications in computer science, esp. Number theory structures, examples, and problems titu. Divisibility and the division algorithm mathematics. 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. Divisibility theory mathematical exercises bioprofe. For example, the numbers 20, 36, and 45 are relatively prime. I recommend this friendly volume for students looking for challenging problems in number theory and teachers of number theory for undergraduates. A divisibility rule is a heuristic for determining whether a positive integer can be evenly divided by another i. 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. A number is divisible by 3 if the sum of its digits is divisible by 3. 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.
Thus in the next definition, d, n, and k are integers. Divisibility number theory is concerned with the properties of the integers. Number theory divisibility tests number theory contents see also tables of. The pace is fast, but the exposition in each chapter is detailed.
Several examples are given and there is an indepth discussion. Elementary number theory divisibility mathematics stack. We now present some examples in which we use the principle of induction. 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. The notation is read a divides b, which is a statement. Number theory explore the powers of divisibility, modular arithmetic, and infinity.
Find rules for divisibility for numbers where the divisor is 2, 4, 8, 5, or 25. I structures, examples, and problems 1 divisibility 15 1. Use the definition of divisibility to show that given any integers a, b, and c, where a. If you dont know the new number s divisibility, you can apply the rule again. Search within a range of numbers put between two numbers. This video gives a geometric interpretation of divisibility as well as the formal definition. These books are both excellent sources of examples, additional practice. Number theorists study prime numbers as well as the properties of. Download for offline reading, highlight, bookmark or take notes while you read number theory. 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.
German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. The book covers the usual topics of introductory number theory. Divisibility two useful properties of divisibility are one of the most useful properties of divisibility is that it is transitive. The reason is students who are new to the topic usually start with problems involving summations followed by. We say that a divides b write a jb i there exists c 2z such that b ac. In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to. God made the integers, all else is the work of man.
Multiple divisibility rules applied to the same number in this way can help quickly determine its prime factorization without. Search for wildcards or unknown words put a in your word or phrase where you want to leave a placeholder. Divisibility, the fundamental theorem of number theory. The true nature of number theory emerges from the first definition. Divisibility by 6 it must be even and divisible by 3. Important concepts are presented with an example, which may also emphasize an application. 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. Introduction to number theory lecture notes university of san. We say that a divides b if there is an integer k such that ak b. Starting with the essentials, the text covers divisibility, powers of integers. The notation means that a divides b for example, since. Here are a few examples of number theoretic problems that have been solved only recently or still remain open.
Structures, examples, and problems ebook written by titu andreescu, dorin andrica. If one number divides a second and the second number divides a third, then the first number divides the third. Accordingly the word number, loosely used or symbolically denoted throughout this book, will be understood an integer, unless otherwise stated. In this book, all numbers are integers, unless specified otherwise.
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