Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Theorem prime number theorem the ratio of the number of primes not exceeding x and xlnx approaches 1 as x grows without bound. Divisibility the notion of divisibility is the central concept of one of the most beautiful subjects in advanced mathematics. A natural number is divisible by 6 if it is divisible by both 2 and 3. You can divide 6 into equal parts of 1, 2, 3, or 6 but not 4 or 5. The properties in the next proposition are easy consequences of the definition of divisibility. Divisible by means when you divide one number by another the result is a whole number examples. This is something that i always forget to teach and it is so useful. Any number divided by 1 will give the number itself, irrespective of how large the number is. If the sum is divisible by 3, then so is the number. Divisibility rules, or divisibility tests, have a wide range of applications in mathematics finding factors, determining if a number is prime or composite, simplifying fractions, probability, etc. The theorem tells us that the number of primes not exceeding x. Being familiar with divisibility and the division algorithm helps us to understand division even more than we already do. Elementary number theory a revision by jim hefferon, st michaels college, 2003dec.
A natural number p is called a prime number if it has exactly two distinct natural number divisors, itself and 1. Sixth grade lesson divisibility rules betterlesson. 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. Discrete mathematics introduction to number theory divisibility example. Results in number theory discovered hundreds of years ago by fermat and euler fuel the modern cryptography keeping your texts, emails, and other electronic data safe. Knowing the rules for divisibility will help you work more efficiently. To see if x is divisible by 7, double the last digit of x and subtract that from the result from the number formed by taking all but the last digit x.
Integers, division, and divisibility calvin university. Divisibility in this book, all numbers are integers, unless speci. 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. Number theory or, in older usage, arithmetic is a branch of. Discrete mathematics introduction to number theory 119. Every time you buy a book from amazon, check your grades on websis, or use a paypal account, you are relying on number theoretic algorithms. For example, 3 is divisible by 1 and 3000 is also divisible by 1 completely. Questions of divisibility, use of the euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, ititi f ft b dinvestigation of perfect numbers and congruences.
Number theoryelementary divisibility wikibooks, open. Factors, factorials, and divisibility sample gmat number theory question duration. Tes global ltd is registered in england company no 02017289 with its registered office at 26 red lion square london wc1r 4hq. The notes contain a useful introduction to important topics that need to be addressed in a course in number theory.
Day 7 number theory 1 number theory in elementary number theory, integers are studied without use of techniques from other mathematical fields. Find rules for divisibility for numbers where the divisor is 2, 4, 8, 5, or 25. Jul 28, 2015 this website and its content is subject to our terms and conditions. The subclassifications of the natural number are given below. Number theory is the study of the set of positive whole numbers. Divisibility rules from 1 to division rules in maths. A number, a, is divisible by a number, b, when b divides into a evenly. Every time you buy a book from amazon, check your grades.
What is the least number of marbles that can satisfy the following situation. The students who are appearing for various competitive examinations where mathematics is on for testing shall also find it useful. Number theoryelementary divisibility wikibooks, open books. A prime number is an integer greater than 1 whose only positive divisors are itself and 1. Divisibility rulesprime or composite numbersmultiplesfactorsprime factorizationleast common multiplegreatest common factordistan. Divisibility by 4 the number formed by its last two digits must be divisible by 4.
In the remainder of the book we will concern ourselves principally with integers. Divisibility by 7 when the units digit is doubled and subtracted from the number. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. 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. The following theorems illustrate a number of important properties of divisibility. Number theory i massachusetts institute of technology.
The first eleven such numbers are 2, 3, 5, 7, 11, , 17, 19, 23, 29, and 31. Note that the number 1 is generally not considered a prime number even though it has no divisors other than itself. These notes serve as course notes for an undergraduate course in number theory. The prime number theorem gives an asymptotic estimate for the number of primes not exceeding x. We proceed to define the con cept of divisibility and the division algorithm. This work presents the discovery and proof of my theorem general case division theorem, which gives methods of determining rules for the even divisibility of natural numbers for a given natural number. This book is designed to meet the needs of the first course in number theory for the undergraduate students of various indian and foreign universities basic number theory by sb malik pdf download. It would be nice if we could solve all these silly water jug questions at once. The first eleven such numbers are 2, 3, 5, 7, 11, 17, 19, 23, 29, and 31. Ppt number theory and cryptography powerpoint presentation. Divisibility rule for 1 doesnt have any particular condition. Prime numbers number theory is the mathematical study of the natural numbers, the positive whole numbers such as 2, 17, and 123. Youll also find out how modular arithmetic is used to help prevent errors in identification numbers, such as the international standard book.
Divisibility by 6 it must be even and divisible by 3. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. His book the arithmetica is a collection of workedout problems where. Considering the remainder modulo an integer is a powerful, foundational tool in number theory. But avoid asking for help, clarification, or responding to other answers. Divisibility, the fundamental theorem of number theory. Despite their ubiquity and apparent simplicity, the natural integers are chockfull of beautiful ideas and open problems. Number theory, known to gauss as arithmetic, studies the properties of the. These ambiguities can be a real source of confusion.
To eliminate this issue, i have the students draw a box around the last 2 digits to get them looking at the number as a whole, not as parts. The systematic study of number theory was initiated around 300b. The number 1 is a special case which is considered neither prime nor composite the number 2 is also special, it is the only even prime. For example, here are some problems in number theory that remain unsolved. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. Introduction to number theory division divisors examples divisibility. The true nature of number theory emerges from the first definition. Discrete mathematics introduction to number theory.
In number theory, the numbers are classified into different types, such as natural numbers, whole numbers, complex numbers, and so on. In a book he was reading around 1630, fermat claimed to have a proof, but not enough space in the margin to write it down. A primary focus of number theory is the study of prime numbers, which can be. Determine whether the following numbers are divisible by 6 or not. It is also possible that a number that doesnt look like an integer is, in fact, an integer e. There are an infinite number of primes, however, as will be proven below. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. Thanks for contributing an answer to mathematics stack exchange. Students struggle with the divisibility rule for 4, looking at the last two digits. I have put some useful rules of divisibility, followed by a worksheet for pupils to practice the rules, followed by answers. To use sets of numbers to find and describe number patterns.