Introduction Integers. Discrete Mathematics Andrei Bulatov

Size: px
Start display at page:

Download "Introduction Integers. Discrete Mathematics Andrei Bulatov"

Transcription

1 Introduction Integers Discrete Mathematics Andrei Bulatov

2 Discrete Mathematics - Integers 9- Integers God made the integers; all else is the work of man Leopold Kroenecker

3 Discrete Mathematics - Integers 9-3 Division Most of useful properties of integers are related to division If a and b are integers with a, we say that a divides b if there is an integer c such that b = ac. When a divides b we say that a is a divisor (factor) of b, and that b is a multiple of a. The notation a b denotes that a divides b. We write a b when a does not divide b Example. Let n and d be positive integers. How many positive integers not exceeding n are divisible by d? The numbers in question have the form dk, where k is a positive integer and < dk n. Therefore, < k n/d. Thus the answer is n/d

4 Discrete Mathematics - Integers 9-4 Properties of Divisibility Let a, b, and c be integers. Then (i) if a b and a c, then a (b + c); (ii) if a b, then a bc for all integers c; (iii) if a b and b c, then a c. Proof. (i) Suppose a b and a c. This means that there are k and m such that b = ak and c = am. Then b + c = ak + am = a(k + m), and a divides b + c.

5 Discrete Mathematics - Integers 9-5 Properties of Divisibility (cntd) If a, b, and c are integers such that a b and a c, then a mb + nc whenever m and n are integers. Proof. By part (ii) it follows that a mb and a nc. By part (i) it follows that a mb + nc. If a b and b a, then a = ±b. Proof. Suppose that a b and b a. Then b = ak and a = bm for some integers k and m. Therefore a = bm = akm, which is possible only if k,m = ±.

6 Discrete Mathematics - Integers 9-6 The Division Algorithm Theorem (The division algorithm) Let a be an integer and d a positive integer. Then there are unique integers q and r, with r < d, such that a = dq + r d is called the divisor, a is called the dividend, q is called the quotient, and r is called the remainder Examples: - Let a = and d = Then = Let a = - and d = 3 Then - = 3 (-4) + - Let a = 3 and d = Then 3 = + 3

7 Discrete Mathematics Primes 3-7 Representation of Integers In most cases we use decimal representation of integers. For example, 657 means = Let b be a positive integer greater than. Then if n is a positive integer, it can be expressed uniquely in the form k k k k n = a b + a b + L+ a b + a where k is a nonnegative number, a k, ak, K,a, a are nonnegative integers less than b, and a k Such a representation of n is called the base b expansion of n, denoted by (a a Ka a ) k k b

8 Discrete Mathematics Primes 3-8 Binary Expansion a Important case of a base is. The base expansion is called the binary expansion of a number k k n = a + a + L+ a + a Find the binary expansion of a k = = = + k 4 a a a 3 4 a5 a6 = + = = + 65 = () 5 5 = + = + a 7

9 Discrete Mathematics Primes 3-9 Hexadecimal Expansion Using A, B, C, D, E, F for,,, 3, 4, 5, respectively, find the base 6 expansion of 7567

10 Discrete Mathematics Primes 3- Primes Every integer n (except for and -) has at least positive divisors, and n (or -n). A positive number that does not have any other positive divisor is called prime Prime numbers:, 3, 5, 7,, 3, 7, 9, 3, 9, 3, 37, 4, 43, Mersenne numbers are the numbers of the form M = n n There are many prime numbers among Mersenne numbers. The greatest known prime number is M = The next candidate is = M M6 A positive number that is not prime is called composite

11 Discrete Mathematics Primes 3- Composite Numbers Every composite number has a prime divisor. Proof. Let S be the set of all composite numbers that do not have a prime divisor Since S N, by the Well-Ordering Principle, it has a least element r. As r is not prime, it has a divisor, therefore, r = uv for some positive integers u and v. u < r and v < r. Therefore u S, and u has a prime divisor p. Since p u and u r, we conclude that p r, a contradiction.

12 Discrete Mathematics Primes 3- How many prime numbers are there? Theorem (Euclid) There are infinitely many prime numbers. Proof. By contradiction. Suppose that { p,p, K, pk} is the set of all prime numbers, and let a = pp Kpk + Since a is greater than any member of the list, a is composite. By the previous statement, a has a prime divisor, that is for some we have p j a Since p j a and p p p Kp we have p a p p Kp = j k A contradiction. n If n is a positive integer, then there are approximately prime ln n numbers not exceeding n j k p j

13 Discrete Mathematics Primes 3-3 Open Problems about Primes Goldbach s Conjecture Every positive even number can be represented as the sum of two prime numbers. For example: 4 = +, 8 = 5 + 3, 4 = Goldbach s conjecture is known to be true for even numbers 7 up to The Twin Prime Conjecture Twin primes are primes that differ by, such as 3 and 5, 5 and 7, and 3, etc. The Twin Prime Conjecture asserts that there are infinitely many twin primes. 7,96 The record twin primes: 6,896,987,339,975 ±

14 Discrete Mathematics Primes 3-4 Homework Exercises from the Book No., 4,, 5, 6 (page 3)

With Question/Answer Animations. Chapter 4

With Question/Answer Animations. Chapter 4 With Question/Answer Animations Chapter 4 Chapter Motivation Number theory is the part of mathematics devoted to the study of the integers and their properties. Key ideas in number theory include divisibility

More information

3 The fundamentals: Algorithms, the integers, and matrices

3 The fundamentals: Algorithms, the integers, and matrices 3 The fundamentals: Algorithms, the integers, and matrices 3.4 The integers and division This section introduces the basics of number theory number theory is the part of mathematics involving integers

More information

4 Number Theory and Cryptography

4 Number Theory and Cryptography 4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic This section introduces the basics of number theory number theory is the part of mathematics involving integers and their properties.

More information

Introduction to Sets and Logic (MATH 1190)

Introduction to Sets and Logic (MATH 1190) Introduction to Sets and Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics and Statistics York University Nov 13, 2014 Quiz announcement The second quiz will be held on Thursday,

More information

not to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results

not to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division

More information

Ch 4.2 Divisibility Properties

Ch 4.2 Divisibility Properties Ch 4.2 Divisibility Properties - Prime numbers and composite numbers - Procedure for determining whether or not a positive integer is a prime - GCF: procedure for finding gcf (Euclidean Algorithm) - Definition:

More information

Math.3336: Discrete Mathematics. Primes and Greatest Common Divisors

Math.3336: Discrete Mathematics. Primes and Greatest Common Divisors Math.3336: Discrete Mathematics Primes and Greatest Common Divisors Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu

More information

Discrete Structures Lecture Primes and Greatest Common Divisor

Discrete Structures Lecture Primes and Greatest Common Divisor DEFINITION 1 EXAMPLE 1.1 EXAMPLE 1.2 An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite.

More information

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary

More information

Math.3336: Discrete Mathematics. Primes and Greatest Common Divisors

Math.3336: Discrete Mathematics. Primes and Greatest Common Divisors Math.3336: Discrete Mathematics Primes and Greatest Common Divisors Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu

More information

MATH10040: Numbers and Functions Homework 1: Solutions

MATH10040: Numbers and Functions Homework 1: Solutions MATH10040: Numbers and Functions Homework 1: Solutions 1. Prove that a Z and if 3 divides into a then 3 divides a. Solution: The statement to be proved is equivalent to the statement: For any a N, if 3

More information

The set of integers will be denoted by Z = {, -3, -2, -1, 0, 1, 2, 3, 4, }

The set of integers will be denoted by Z = {, -3, -2, -1, 0, 1, 2, 3, 4, } Integers and Division 1 The Integers and Division This area of discrete mathematics belongs to the area of Number Theory. Some applications of the concepts in this section include generating pseudorandom

More information

Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006

Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006 Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking

More information

MATH 215 Final. M4. For all a, b in Z, a b = b a.

MATH 215 Final. M4. For all a, b in Z, a b = b a. MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on

More information

Homework 3, solutions

Homework 3, solutions Homework 3, solutions Problem 1. Read the proof of Proposition 1.22 (page 32) in the book. Using simialr method prove that there are infinitely many prime numbers of the form 3n 2. Solution. Note that

More information

SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION

SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.4 Strong Mathematical Induction and the Well-Ordering Principle for the Integers Copyright

More information

8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

More information

Exercises Exercises. 2. Determine whether each of these integers is prime. a) 21. b) 29. c) 71. d) 97. e) 111. f) 143. a) 19. b) 27. c) 93.

Exercises Exercises. 2. Determine whether each of these integers is prime. a) 21. b) 29. c) 71. d) 97. e) 111. f) 143. a) 19. b) 27. c) 93. Exercises Exercises 1. Determine whether each of these integers is prime. a) 21 b) 29 c) 71 d) 97 e) 111 f) 143 2. Determine whether each of these integers is prime. a) 19 b) 27 c) 93 d) 101 e) 107 f)

More information

REAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.

REAL NUMBERS. Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. REAL NUMBERS Introduction Euclid s Division Algorithm Any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Fundamental

More information

Solutions to Assignment 1

Solutions to Assignment 1 Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive

More information

An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.

An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1

More information

Math 3336 Section 4.3 Primes and Greatest Common Divisors. Prime Numbers and their Properties Conjectures and Open Problems About Primes

Math 3336 Section 4.3 Primes and Greatest Common Divisors. Prime Numbers and their Properties Conjectures and Open Problems About Primes Math 3336 Section 4.3 Primes and Greatest Common Divisors Prime Numbers and their Properties Conjectures and Open Problems About Primes Greatest Common Divisors and Least Common Multiples The Euclidian

More information

Introduction to Number Theory. The study of the integers

Introduction to Number Theory. The study of the integers Introduction to Number Theory The study of the integers of Integers, The set of integers = {... 3, 2, 1, 0, 1, 2, 3,...}. In this lecture, if nothing is said about a variable, it is an integer. Def. We

More information

We want to show P (n) is true for all integers

We want to show P (n) is true for all integers Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to

More information

A number that can be written as, where p and q are integers and q Number.

A number that can be written as, where p and q are integers and q Number. RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.

More information

Appendix: a brief history of numbers

Appendix: a brief history of numbers Appendix: a brief history of numbers God created the natural numbers. Everything else is the work of man. Leopold Kronecker (1823 1891) Fundamentals of Computing 2017 18 (2, appendix) http://www.dcs.bbk.ac.uk/~michael/foc/foc.html

More information

. As the binomial coefficients are integers we have that. 2 n(n 1).

. As the binomial coefficients are integers we have that. 2 n(n 1). Math 580 Homework. 1. Divisibility. Definition 1. Let a, b be integers with a 0. Then b divides b iff there is an integer k such that b = ka. In the case we write a b. In this case we also say a is a factor

More information

Downloaded from

Downloaded from Topic : Real Numbers Class : X Concepts 1. Euclid's Division Lemma 2. Euclid's Division Algorithm 3. Prime Factorization 4. Fundamental Theorem of Arithmetic 5. Decimal expansion of rational numbers A

More information

MATHEMATICS X l Let x = p q be a rational number, such l If p, q, r are any three positive integers, then, l that the prime factorisation of q is of t

MATHEMATICS X l Let x = p q be a rational number, such l If p, q, r are any three positive integers, then, l that the prime factorisation of q is of t CHAPTER 1 Real Numbers [N.C.E.R.T. Chapter 1] POINTS FOR QUICK REVISION l Euclid s Division Lemma: Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r

More information

Summary Slides for MATH 342 June 25, 2018

Summary Slides for MATH 342 June 25, 2018 Summary Slides for MATH 342 June 25, 2018 Summary slides based on Elementary Number Theory and its applications by Kenneth Rosen and The Theory of Numbers by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery.

More information

Chapter 1 A Survey of Divisibility 14

Chapter 1 A Survey of Divisibility 14 Chapter 1 A Survey of Divisibility 14 SECTION C Euclidean Algorithm By the end of this section you will be able to use properties of the greatest common divisor (gcd) obtain the gcd using the Euclidean

More information

7. Prime Numbers Part VI of PJE

7. Prime Numbers Part VI of PJE 7. Prime Numbers Part VI of PJE 7.1 Definition (p.277) A positive integer n is prime when n > 1 and the only divisors are ±1 and +n. That is D (n) = { n 1 1 n}. Otherwise n > 1 is said to be composite.

More information

NOTES ON SIMPLE NUMBER THEORY

NOTES ON SIMPLE NUMBER THEORY NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,

More information

Wednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory).

Wednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory). Wednesday, February 21 Today we will begin Course Notes Chapter 5 (Number Theory). 1 Return to Chapter 5 In discussing Methods of Proof (Chapter 3, Section 2) we introduced the divisibility relation from

More information

Lecture 4: Number theory

Lecture 4: Number theory Lecture 4: Number theory Rajat Mittal IIT Kanpur In the next few classes we will talk about the basics of number theory. Number theory studies the properties of natural numbers and is considered one of

More information

Section Summary. Division Division Algorithm Modular Arithmetic

Section Summary. Division Division Algorithm Modular Arithmetic 1 Chapter Motivation Number theory is the part of mathematics devoted to the study of the integers and their properties. Key ideas in number theory include divisibility and the primality of integers. Representations

More information

EDULABZ INTERNATIONAL NUMBER SYSTEM

EDULABZ INTERNATIONAL NUMBER SYSTEM NUMBER SYSTEM 1. Find the product of the place value of 8 and the face value of 7 in the number 7801. Ans. Place value of 8 in 7801 = 800, Face value of 7 in 7801 = 7 Required product = 800 7 = 00. How

More information

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. (a) Circle the prime

More information

CHAPTER 6. Prime Numbers. Definition and Fundamental Results

CHAPTER 6. Prime Numbers. Definition and Fundamental Results CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n

More information

CISC 1400 Discrete Structures

CISC 1400 Discrete Structures CISC 1400 Discrete Structures Chapter 2 Sequences What is Sequence? A sequence is an ordered list of objects or elements. For example, 1, 2, 3, 4, 5, 6, 7, 8 Each object/element is called a term. 1 st

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Representation of integers Mathematical Induction Reading (Epp s textbook) 5.1 5.3 1 Representations of Integers Let b be a positive integer greater than 1. Then if n is a positive

More information

Bell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.

Bell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible. Bell Quiz 2-3 2 pts Determine the end behavior of the graph using limit notation. 5 2 1. g( ) = 8 + 13 7 3 pts Find a function with the given zeros. 4. -1, 2 5 pts possible Ch 2A Big Ideas 1 Questions

More information

Standard forms for writing numbers

Standard forms for writing numbers Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,

More information

4.3 - Primes and Greatest Common Divisors

4.3 - Primes and Greatest Common Divisors 4.3 - Primes and Greatest Common Divisors Introduction We focus on properties of integers and prime factors Primes Definition 1 An integer p greater than 1 is called prime if the only positive factors

More information

Discrete Mathematics. Spring 2017

Discrete Mathematics. Spring 2017 Discrete Mathematics Spring 2017 Previous Lecture Principle of Mathematical Induction Mathematical Induction: rule of inference Mathematical Induction: Conjecturing and Proving Climbing an Infinite Ladder

More information

Elementary Number Theory

Elementary Number Theory Elementary Number Theory A revision by Jim Hefferon, St Michael s College, 2003-Dec of notes by W. Edwin Clark, University of South Florida, 2002-Dec L A TEX source compiled on January 5, 2004 by Jim Hefferon,

More information

Complete Induction and the Well- Ordering Principle

Complete Induction and the Well- Ordering Principle Complete Induction and the Well- Ordering Principle Complete Induction as a Rule of Inference In mathematical proofs, complete induction (PCI) is a rule of inference of the form P (a) P (a + 1) P (b) k

More information

Number Theory and Graph Theory

Number Theory and Graph Theory 1 Number Theory and Graph Theory Chapter 1 Introduction and Divisibility By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com 1 DIVISION

More information

Mat Week 8. Week 8. gcd() Mat Bases. Integers & Computers. Linear Combos. Week 8. Induction Proofs. Fall 2013

Mat Week 8. Week 8. gcd() Mat Bases. Integers & Computers. Linear Combos. Week 8. Induction Proofs. Fall 2013 Fall 2013 Student Responsibilities Reading: Textbook, Section 3.7, 4.1, & 5.2 Assignments: Sections 3.6, 3.7, 4.1 Proof Worksheets Attendance: Strongly Encouraged Overview 3.6 Integers and Algorithms 3.7

More information

Number Theory and Cryptography

Number Theory and Cryptography . All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Number Theory and

More information

Student Responsibilities Week 8. Mat Section 3.6 Integers and Algorithms. Algorithm to Find gcd()

Student Responsibilities Week 8. Mat Section 3.6 Integers and Algorithms. Algorithm to Find gcd() Student Responsibilities Week 8 Mat 2345 Week 8 Reading: Textbook, Section 3.7, 4.1, & 5.2 Assignments: Sections 3.6, 3.7, 4.1 Induction Proof Worksheets Attendance: Strongly Encouraged Fall 2013 Week

More information

CSE 215: Foundations of Computer Science Recitation Exercises Set #5 Stony Brook University. Name: ID#: Section #: Score: / 4

CSE 215: Foundations of Computer Science Recitation Exercises Set #5 Stony Brook University. Name: ID#: Section #: Score: / 4 CSE 215: Foundations of Computer Science Recitation Exercises Set #5 Stony Brook University Name: ID#: Section #: Score: / 4 Unit 10: Proofs by Contradiction and Contraposition 1. Prove the following statement

More information

4. Number Theory (Part 2)

4. Number Theory (Part 2) 4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.

More information

Ma/CS 6a Class 2: Congruences

Ma/CS 6a Class 2: Congruences Ma/CS 6a Class 2: Congruences 1 + 1 5 (mod 3) By Adam Sheffer Reminder: Public Key Cryptography Idea. Use a public key which is used for encryption and a private key used for decryption. Alice encrypts

More information

Algorithmic number theory. Questions/Complaints About Homework? The division algorithm. Division

Algorithmic number theory. Questions/Complaints About Homework? The division algorithm. Division Questions/Complaints About Homework? Here s the procedure for homework questions/complaints: 1. Read the solutions first. 2. Talk to the person who graded it (check initials) 3. If (1) and (2) don t work,

More information

A Readable Introduction to Real Mathematics

A Readable Introduction to Real Mathematics Solutions to selected problems in the book A Readable Introduction to Real Mathematics D. Rosenthal, D. Rosenthal, P. Rosenthal Chapter 7: The Euclidean Algorithm and Applications 1. Find the greatest

More information

Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively

Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively 6 Prime Numbers Part VI of PJE 6.1 Fundamental Results Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively D (p) = { p 1 1 p}. Otherwise

More information

3 The Fundamentals:Algorithms, the Integers, and Matrices

3 The Fundamentals:Algorithms, the Integers, and Matrices 3 The Fundamentals:Algorithms, the Integers, and Matrices 3.1 Algrithms Definition 1 An algorithm is a finite set of instructions for performing a computation or solving a problem. Dijkstra s mini language

More information

Primes. Rational, Gaussian, Industrial Strength, etc. Robert Campbell 11/29/2010 1

Primes. Rational, Gaussian, Industrial Strength, etc. Robert Campbell 11/29/2010 1 Primes Rational, Gaussian, Industrial Strength, etc Robert Campbell 11/29/2010 1 Primes and Theory Number Theory to Abstract Algebra History Euclid to Wiles Computation pencil to supercomputer Practical

More information

Chapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices. Integers & Algorithms (2.5)

Chapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices. Integers & Algorithms (2.5) CSE 54 Discrete Mathematics & Chapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices Integers & Algorithms (Section 2.5) by Kenneth H. Rosen, Discrete Mathematics & its Applications,

More information

Proofs. Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm. Reading (Epp s textbook)

Proofs. Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm. Reading (Epp s textbook) Proofs Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm Reading (Epp s textbook) 4.3 4.8 1 Divisibility The notation d n is read d divides n. Symbolically,

More information

Math 412: Number Theory Lecture 3: Prime Decomposition of

Math 412: Number Theory Lecture 3: Prime Decomposition of Math 412: Number Theory Lecture 3: Prime Decomposition of Integers Gexin Yu gyu@wm.edu College of William and Mary Prime numbers Definition: a (positive) integer p is prime if p has no divisor other than

More information

Warm-Up. Use long division to divide 5 into

Warm-Up. Use long division to divide 5 into Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use

More information

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number

More information

Discrete Mathematics GCD, LCM, RSA Algorithm

Discrete Mathematics GCD, LCM, RSA Algorithm Discrete Mathematics GCD, LCM, RSA Algorithm Abdul Hameed http://informationtechnology.pk/pucit abdul.hameed@pucit.edu.pk Lecture 16 Greatest Common Divisor 2 Greatest common divisor The greatest common

More information

Four Basic Sets. Divisors

Four Basic Sets. Divisors Four Basic Sets Z = the integers Q = the rationals R = the real numbers C = the complex numbers Divisors Definition. Suppose a 0 and b = ax, where a, b, and x are integers. Then we say a divides b (or

More information

The Real Number System

The Real Number System MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely

More information

An Algorithm for Prime Factorization

An Algorithm for Prime Factorization An Algorithm for Prime Factorization Fact: If a is the smallest number > 1 that divides n, then a is prime. Proof: By contradiction. (Left to the reader.) A multiset is like a set, except repetitions are

More information

Elementary Number Theory

Elementary Number Theory Elementary Number Theory CIS002-2 Computational Alegrba and Number Theory David Goodwin david.goodwin@perisic.com 09:00, Tuesday 25 th October 2011 Contents 1 Some definitions 2 Divisibility Divisors Euclid

More information

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a

More information

Primes and Factorization

Primes and Factorization Primes and Factorization 1 A prime number is an integer greater than 1 with no proper divisors. The list begins 2, 3, 5, 7, 11, 13, 19,... See http://primes.utm.edu/ for a wealth of information about primes.

More information

Numbers and their divisors

Numbers and their divisors Chapter 1 Numbers and their divisors 1.1 Some number theoretic functions Theorem 1.1 (Fundamental Theorem of Arithmetic). Every positive integer > 1 is uniquely the product of distinct prime powers: n

More information

Deepening Mathematics Instruction for Secondary Teachers: Algebraic Structures

Deepening Mathematics Instruction for Secondary Teachers: Algebraic Structures Deepening Mathematics Instruction for Secondary Teachers: Algebraic Structures Lance Burger Fresno State Preliminary Edition Contents Preface ix 1 Z The Integers 1 1.1 What are the Integers?......................

More information

LECTURE 1: DIVISIBILITY. 1. Introduction Number theory concerns itself with studying the multiplicative and additive structure of the natural numbers

LECTURE 1: DIVISIBILITY. 1. Introduction Number theory concerns itself with studying the multiplicative and additive structure of the natural numbers LECTURE 1: DIVISIBILITY 1. Introduction Number theory concerns itself with studying the multiplicative and additive structure of the natural numbers N = {1, 2, 3,... }. Frequently, number theoretic questions

More information

CS March 17, 2009

CS March 17, 2009 Discrete Mathematics CS 2610 March 17, 2009 Number Theory Elementary number theory, concerned with numbers, usually integers and their properties or rational numbers mainly divisibility among integers

More information

Fundamentals of Pure Mathematics - Problem Sheet

Fundamentals of Pure Mathematics - Problem Sheet Fundamentals of Pure Mathematics - Problem Sheet ( ) = Straightforward but illustrates a basic idea (*) = Harder Note: R, Z denote the real numbers, integers, etc. assumed to be real numbers. In questions

More information

cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska

cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska LECTURE 12 CHAPTER 4 NUMBER THEORY PART1: Divisibility PART 2: Primes PART 1: DIVISIBILITY Basic Definitions Definition Given m,n Z, we say

More information

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS

ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems Factor each polynomial completely using the given factor and long division. 3. x 3 + 3x 2 18x 40; x 4 So, x 3 + 3x 2 18x 40 = (x 4)(x 2 + 7x + 10). Factoring the quadratic expression yields x 3 + 3x 2

More information

Math /Foundations of Algebra/Fall 2017 Foundations of the Foundations: Proofs

Math /Foundations of Algebra/Fall 2017 Foundations of the Foundations: Proofs Math 4030-001/Foundations of Algebra/Fall 017 Foundations of the Foundations: Proofs A proof is a demonstration of the truth of a mathematical statement. We already know what a mathematical statement is.

More information

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element. The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring

More information

Here is another characterization of prime numbers.

Here is another characterization of prime numbers. Here is another characterization of prime numbers. Theorem p is prime it has no divisors d that satisfy < d p. Proof [ ] If p is prime then it has no divisors d that satisfy < d < p, so clearly no divisor

More information

God may not play dice with the universe, but something strange is going on with the prime numbers.

God may not play dice with the universe, but something strange is going on with the prime numbers. Primes: Definitions God may not play dice with the universe, but something strange is going on with the prime numbers. P. Erdös (attributed by Carl Pomerance) Def: A prime integer is a number whose only

More information

Real Number. Euclid s division algorithm is based on the above lemma.

Real Number. Euclid s division algorithm is based on the above lemma. 1 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius Basic Concepts 1. Euclid s division lemma Given two positive integers a and b, there exist unique integers q and r

More information

Downloaded from

Downloaded from Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page

More information

To hand in: (a) Prove that a group G is abelian (= commutative) if and only if (xy) 2 = x 2 y 2 for all x, y G.

To hand in: (a) Prove that a group G is abelian (= commutative) if and only if (xy) 2 = x 2 y 2 for all x, y G. Homework #6. Due Thursday, October 14th Reading: For this homework assignment: Sections 3.3 and 3.4 (up to page 167) Before the class next Thursday: Sections 3.5 and 3.4 (pp. 168-171). Also review the

More information

This is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time.

This is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time. 8 Modular Arithmetic We introduce an operator mod. Let d be a positive integer. For c a nonnegative integer, the value c mod d is the remainder when c is divided by d. For example, c mod d = 0 if and only

More information

D i v i s i b i l i t y

D i v i s i b i l i t y a D i v i s i b i l i t y statement which asserts that all numbers possess a certain property cannot be proved in this manner. The assertion, "Every prime number of the form An + 1 is a sum of two squares,"

More information

Numbers. 2.1 Integers. P(n) = n(n 4 5n 2 + 4) = n(n 2 1)(n 2 4) = (n 2)(n 1)n(n + 1)(n + 2); 120 =

Numbers. 2.1 Integers. P(n) = n(n 4 5n 2 + 4) = n(n 2 1)(n 2 4) = (n 2)(n 1)n(n + 1)(n + 2); 120 = 2 Numbers 2.1 Integers You remember the definition of a prime number. On p. 7, we defined a prime number and formulated the Fundamental Theorem of Arithmetic. Numerous beautiful results can be presented

More information

COT 3100 Applications of Discrete Structures Dr. Michael P. Frank

COT 3100 Applications of Discrete Structures Dr. Michael P. Frank University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank Slides for a Course Based on the Text Discrete Mathematics

More information

Multiplying and dividing integers Student Activity Sheet 2; use with Exploring Patterns of multiplying and dividing integers

Multiplying and dividing integers Student Activity Sheet 2; use with Exploring Patterns of multiplying and dividing integers 1. Study the multiplication problems in the tables. [EX2, page 1] Factors Product Factors Product (4)(3) 12 ( 4)(3) 12 (4)(2) 8 ( 4)(2) 8 (4)(1) 4 ( 4)(1) 4 (4)(0) 0 ( 4)(0) 0 (4)( 1) 4 ( 4)( 1) 4 (4)(

More information

Chapter 2. Divisibility. 2.1 Common Divisors

Chapter 2. Divisibility. 2.1 Common Divisors Chapter 2 Divisibility 2.1 Common Divisors Definition 2.1.1. Let a and b be integers. A common divisor of a and b is any integer that divides both a and b. Suppose that a and b are not both zero. By Proposition

More information

Around Prime Numbers And Twin Primes

Around Prime Numbers And Twin Primes Theoretical Mathematics & Alications, vol. 3, no. 1, 2013, 211-220 ISSN: 1792-9687 (rint), 1792-9709 (online) Scienress Ltd, 2013 Around Prime Numbers And Twin Primes Ikorong Anouk Gilbert Nemron 1 Abstract

More information

CSE 20 DISCRETE MATH. Winter

CSE 20 DISCRETE MATH. Winter CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Define and use the congruence modulo m equivalence relation Perform computations using modular arithmetic

More information

Discrete Mathematics. Spring 2017

Discrete Mathematics. Spring 2017 Discrete Mathematics Spring 2017 Previous Lecture Principle of Mathematical Induction Mathematical Induction: Rule of Inference Mathematical Induction: Conjecturing and Proving Mathematical Induction:

More information

Long division for integers

Long division for integers Feasting on Leftovers January 2011 Summary notes on decimal representation of rational numbers Long division for integers Contents 1. Terminology 2. Description of division algorithm for integers (optional

More information

Topics in Number Theory

Topics in Number Theory Topics in Number Theory Valentin Goranko Topics in Number Theory This essay is an introduction to some basic topics in number theory related to divisibility, prime numbers and congruences. These topics

More information

Lecture 7 Number Theory Euiseong Seo

Lecture 7 Number Theory Euiseong Seo Lecture 7 Number Theory Euiseong Seo (euiseong@skku.edu) 1 Number Theory God created the integers. All else is the work of man Leopold Kronecker Study of the property of the integers Specifically, integer

More information

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}

WORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...} WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not

More information

Well-Ordering Principle. Axiom: Every nonempty subset of Z + has a least element. That is, if S Z + and S, then S has a smallest element.

Well-Ordering Principle. Axiom: Every nonempty subset of Z + has a least element. That is, if S Z + and S, then S has a smallest element. Well-Ordering Principle Axiom: Every nonempty subset of Z + has a least element. That is, if S Z + and S, then S has a smallest element. Well-Ordering Principle Example: Use well-ordering property to prove

More information