To use sets of numbers to find and describe number patterns. Discrete mathematics introduction to number theory. As it turns out, there are a number of interesting computerrelated applications of basic number theory. Divisibility is the property of an integer number to be divided by another, resulting an integer number. Six is such a perfect number, since it is the sum of its parts 1, 2, and 3. 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 a lesson in abstract algebra presented to prof jose binaluyo 2. In this book, all numbers are integers, unless specified otherwise. Sep 26, 2014 divisibility rules properties of divisibility 1.
Browse other questions tagged number theory or ask your own question. Mathematical fun with happy numbers use happy numbers in your math classroom at either elementary or secondary level. Number theoryelementary divisibility wikibooks, open books for. Members of this class represent a rich variety of backgrounds and perspectives. This video is provided by the learning assistance center of howard community college. Divisibility and the division algorithm mathematics.
Basics of divisibility in this chapter, we will discuss the divisibility of integers, the set of integers is denoted by. Elementary properties of divisibility edit divisibility is a key concept in number theory. The notes contain a useful introduction to important topics that need to be addressed in a course in number theory. Edwin clark copyleft means that unrestricted redistribution and modi.
Number theory has always fascinated amateurs as well as professional mathematicians. Divisibility in this book, all numbers are integers, unless speci. Thanks for contributing an answer to mathematics stack exchange. This 1st volume in the series history of the theory of numbers presents the material related to the subjects of divisibility and primality. The number formed by the tens and units digit of the number must be divisible by. Famous theorems of mathematicsnumber theory wikibooks. For example, 3 is divisible by 1 and 3000 is also divisible by 1 completely. Introduction to number theory 1 divisibility semantic scholar. Number theory, known to gauss as arithmetic, studies the properties of the integers. Jun 03, 20 an explanation of divisibility notation and some divisibility theorems.
Divisibility and primality dover books on mathematics. 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. Sometimes called higher arithmetic, it is among the oldest and most natural of mathematical pursuits. Number theoryelementary divisibility wikibooks, open books. 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.
This book, which presupposes familiarity only with the most elementary concepts of arithmetic divisibility properties, greatest common divisor, etc. 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. Any number divided by 1 will give the number itself, irrespective of how large the number is. 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. These rules are collectively called rules of divisibility. Well be examining integer properties in these notes, so well adopt the.
Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Discrete mathematics introduction to number theory 519 divisibility properties, cont. 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. New solutions often require the ingenious use of earlier mathematical. 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. Number theory elementary properties of divisibility.
Paused youre listening to a sample of the audible audio edition. We now discuss the concept of divisibility and its properties. More formally, for a 6 0 we say that divides b if there is another integer k such that b ka. These notes serve as course notes for an undergraduate course in number the ory. The true nature of number theory emerges from the first definition. Considering the remainder modulo an integer is a powerful, foundational tool in number theory. Simple properties of divisibility proofs on page 21. Discrete mathematics introduction to number theory divisibility example. Most of the properties are quite obvious, but it is still a good idea to know how to prove them. In the algebraic number theory, the concept of divisibility will be extended to general algebraic number fields. Shipping may be from multiple locations in the us or from the uk, depending on stock availability. The set z of all integers, which this book is all about, consists of all positive and negative integers.
Number theory concerns the former case, and discovers criteria upon which one can decide about divisibility of two integers. 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. Divisibility some properties of divisibility prime numbers. We are discussing some properties without dealing the proof. In a book he was reading around 1630, fermat claimed to have a proof, but not enough space in the margin to write it. The next theorem records the basic properties of divisibility that are intu itively clear, but easily established from the definition.
Theorem l for all numbers a and b, where b 1 0, there is an integer e and a number. An explanation of divisibility notation and some divisibility theorems. 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. A prime number is an integer greater than 1 whose only positive divisors are itself and 1. Introduction to number theory mathematics libretexts.
Mar 25, 2016 number theory elementary properties of divisibility. For example, here are some problems in number theory that remain unsolved. Discrete mathematics introduction to number theory 419 properties of divisibility i theorem 1. Number theoryelementary divisibility wikibooks, open. Number theory, branch of mathematics concerned with properties of the positive integers 1, 2, 3. The sum of digits of the number must be divisible by 3 3 3.
The number should have 0, 2, 4, 6, 0, \ 2, \ 4, \ 6, 0, 2, 4, 6, or 8 8 8 as the units digit. Number theory elementary properties of divisibility youtube. 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. Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Divisibility rules from 1 to division rules in maths. 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. The above example should convince you that the well known divisibility test for 9 is true. Every time you buy a book from amazon, check your grades on websis, or use a. An introduction to the theory of numbers open textbook library. Divisibility the notion of divisibility is the central concept of one of the most beautiful subjects in advanced mathematics. These ambiguities can be a real source of confusion. Discrete mathematics introduction to number theory 119. The first eleven such numbers are 2, 3, 5, 7, 11, 17, 19, 23, 29, and 31. 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.
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. An introduction to the theory of numbers open textbook. In elementary number theory, integers are studied without use of techniques from other mathematical fields. 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.
In the remainder of the book we will concern ourselves principally with integers. A natural number p is called a prime number if it has exactly two distinct natural number divisors, itself and 1. Solve integer equations, determine remainders of powers, and much more with the power of modular arithmetic. 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. Divisibility, the fundamental theorem of number theory. The properties of divisibility listed here follow easily from the definition. We will give a few detailed proofs of some of the basic facts about divisibility. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. Edwin clark department of mathematics university of south florida revised june 2, 2003 copyleft 2002 by w. The following theorems illustrate a number of important properties of divisibility. Divisibility theory mathematical exercises bioprofe.
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