Number theory elementary properties of divisibility. Divisibility the notion of divisibility is the central concept of one of the most beautiful subjects in advanced mathematics. Divisibility rules from 1 to division rules in maths. Number theory, branch of mathematics concerned with properties of the positive integers 1, 2, 3. Introduction to number theory 1 divisibility semantic scholar. Discrete mathematics introduction to number theory 119. Mar 25, 2016 number theory elementary properties of divisibility.
We say that an integer a \ displaystyle a is divisible by a nonzero integer b \ displaystyle b if there exists an integer c \ displaystyle c such that a b c \ displaystyle abc. In the following chapters on divisibility rules, we shall introduce each of them, and take it to the next level by using algebra in some cases, in addition to arithmetic to investigate why the divisibility rule works. Every time you buy a book from amazon, check your grades on websis, or use a. Theorem l for all numbers a and b, where b 1 0, there is an integer e and a number. Browse other questions tagged number theory or ask your own question. Use the division algorithm to find the quotient and the remainder when 76 is divided by use the division algorithm to find the quotient and the remainder when 100 is divided by.
Any number divided by 1 will give the number itself, irrespective of how large the number is. Every technique is followed by problems as well as detailed hints and solutions that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. A lively introduction with proofs, applications, and stories, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications. The number should have 0, 2, 4, 6, 0, \ 2, \ 4, \ 6, 0, 2, 4, 6, or 8 8 8 as the units digit. In this book, all numbers are integers, unless specified otherwise. A prime number is an integer greater than 1 whose only positive divisors are itself and 1. The above example should convince you that the well known divisibility test for 9 is true. Divisibility theory mathematical exercises bioprofe. Number theory, known to gauss as arithmetic, studies the properties of the integers. Test of divisibility by 11if the digits at odd and even places of a given number are equal or differ by a number divisible by 11, then the given number is divisible by 11. Shipping may be from multiple locations in the us or from the uk, depending on stock availability. An explanation of divisibility notation and some divisibility theorems. Sometimes called higher arithmetic, it is among the oldest and most natural of mathematical pursuits. Divisibility is the property of an integer number to be divided by another, resulting an integer number.
Well be examining integer properties in these notes, so well adopt the. The sum of digits of the number must be divisible by 3 3 3. In the remainder of the book we will concern ourselves principally with integers. Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. This video is provided by the learning assistance center of howard community college. New solutions often require the ingenious use of earlier mathematical. These ambiguities can be a real source of confusion. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. An introduction to the theory of numbers open textbook library. Mathematical fun with happy numbers use happy numbers in your math classroom at either elementary or secondary level. Discrete mathematics introduction to number theory divisibility example.
Famous theorems of mathematicsnumber theory wikibooks. In the algebraic number theory, the concept of divisibility will be extended to general algebraic number fields. We will give a few detailed proofs of some of the basic facts about divisibility. Introduction to number theory mathematics libretexts. To use sets of numbers to find and describe number patterns. Number theoryelementary divisibility wikibooks, open books. Divisibility, the fundamental theorem of number theory. Most of the properties are quite obvious, but it is still a good idea to know how to prove them. This 1st volume in the series history of the theory of numbers presents the material related to the subjects of divisibility and primality. Discrete mathematics introduction to number theory. Paused youre listening to a sample of the audible audio edition. We now discuss the concept of divisibility and its properties.
Thanks for contributing an answer to mathematics stack exchange. 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. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. The true nature of number theory emerges from the first definition. Six is such a perfect number, since it is the sum of its parts 1, 2, and 3. Number theoryelementary divisibility wikibooks, open. The set z of all integers, which this book is all about, consists of all positive and negative integers. Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. 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. Number theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptogra. Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. For example, here are some problems in number theory that remain unsolved. Number theoryelementary divisibility wikibooks, open books for.
The next theorem records the basic properties of divisibility that are intu itively clear, but easily established from the definition. A natural number p is called a prime number if it has exactly two distinct natural number divisors, itself and 1. Edwin clark department of mathematics university of south florida revised june 2, 2003 copyleft 2002 by w. Number theory is the branch of mathematics that deals with integers and their properties, especially properties relating to arithmetic operations like addition, subtraction, multiplication and division.
The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. Discrete mathematics introduction to number theory 419 properties of divisibility i theorem 1. In a book he was reading around 1630, fermat claimed to have a proof, but not enough space in the margin to write it. This book, which presupposes familiarity only with the most elementary concepts of arithmetic divisibility properties, greatest common divisor, etc. For example, 3 is divisible by 1 and 3000 is also divisible by 1 completely.
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 some properties of divisibility prime numbers. These notes serve as course notes for an undergraduate course in number the ory. Sep 26, 2014 divisibility rules properties of divisibility 1. Basics of divisibility in this chapter, we will discuss the divisibility of integers, the set of integers is denoted by. What is the least number of marbles that can satisfy the following situation. Edwin clark copyleft means that unrestricted redistribution and modi. We are discussing some properties without dealing the proof. It is also possible that a number that doesnt look like an integer is, in fact, an integer e. The first eleven such numbers are 2, 3, 5, 7, 11, 17, 19, 23, 29, and 31. These rules are collectively called rules of divisibility. 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. As it turns out, there are a number of interesting computerrelated applications of basic number theory. The properties of divisibility listed here follow easily from the definition.
Divisibility and the division algorithm mathematics. Considering the remainder modulo an integer is a powerful, foundational tool in number theory. Divisibility and primality dover books on mathematics. 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 a lesson in abstract algebra presented to prof jose binaluyo 2. Solve integer equations, determine remainders of powers, and much more with the power of modular arithmetic. In the additive number theory, we will be dealing with the additive properties of prime numbers and with the progress made in solving the goldbach hypothesis. Questions of divisibility, use of the euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. In elementary number theory, integers are studied without use of techniques from other mathematical fields. Elementary properties of divisibility edit divisibility is a key concept in number theory.
Number theory elementary properties of divisibility youtube. Jun 03, 20 an explanation of divisibility notation and some divisibility theorems. Simple properties of divisibility proofs on page 21. Members of this class represent a rich variety of backgrounds and perspectives. Number theory has always fascinated amateurs as well as professional mathematicians. Divisibility rule for 1 doesnt have any particular condition. The following theorems illustrate a number of important properties of divisibility. More formally, for a 6 0 we say that divides b if there is another integer k such that b ka. An introduction to the theory of numbers open textbook. Divisibility in this book, all numbers are integers, unless speci.
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