Elementary Properties of the Integers

Size: px
Start display at page:

Download "Elementary Properties of the Integers"

Transcription

1 Elementary Properties of the Integers 1 1. Basis Representation Theorem (Thm 1-3) 2. Euclid s Division Lemma (Thm 2-1) 3. Greatest Common Divisor 4. Properties of Prime Numbers 5. Fundamental Theorem of Arithmetic (Thm 2-5)

2 The Integers: Z 2 For most of this course, we restrict ourselves to the set of integers. Some helpful hints to keep in mind: Addition, subtraction, and multiplication work normally. The sum/difference/product of two integers is always another integer. Commutative, associative, and distributive laws hold. The reciprocal of an integer is almost never an integer. What are the exceptions? Division/fractions must be handled with care. If you use a fraction within an equation, be ABSOLUTELY CERTAIN that the fraction represents an integer. If you are unsure, don t use the fraction.

3 It is valid to use constant fractions within inequalities (example: x > 1 2 ), but it may be confusing to do so. Consider the sets: A = { x Z : x > 1 }, B = {x Z : x 1}. 2 3 The Cancellation Laws still work. If ax = ay, we can cancel the a from both sides to get x = y, as long as a 0. The same is true for xa = ya. Nonnegative integer exponents work normally. Negative or fractional exponents should generally be avoided. In particular, reciprocals and roots are not always defined, and might behave differently even when they exist.

4 The Natural Numbers: N We will occasionally restrict ourselves to non-negative integers. These are called Natural Numbers (or Whole Numbers, in older texts). Addition and multiplication work normally. Negation does not make sense. Subtraction is not always defined. Reciprocals, division/fractions, exponents/roots have the same issues as in Z. 4 A nonempty subset of N always contains a minimal element. known as Well-Ordering or the Least Integer Principle. This is Mathematical induction can (and often will be) used to prove that a result holds for all natural numbers.

5 Integer Division with Remainder Theorem (Euclid s Division Lemma (Thm 2-1)). For each integer j and each positive integer k, there is a unique pair of integers q and r such that 0 r < k and j = kq + r. For j > 0, this is the familiar division problem j k = q REM r. One way to find the values of q and r: If j < k, we can choose q = 0 and r = j. If j k, then j k 0 is still positive. If j k k, subtract another copy of k (to get j 2k 0). Repeating this, we ll eventually get a j qk that is nonnegative and smaller than k. We let r = j qk. 5

6 Note: This is certainly NOT the most efficient way to compute a quotient/reminder, but it makes for an easy proof (using Well-Ordering). Proof. (Existence): Given j and k, consider the subset of natural numbers S = {j xk : x Z}. Why is this set nonempty? By the Well-Ordering Principle, S has a smallest element. Let r = j qk be this smallest element. We need to show r < k. Assume this is not true, then r k and thus r k 0. However, r k = (j qk) k = j (q + 1)k. 6 Thus r k S, but this is impossible, since r k < r and r was the smallest element of S.

7 (Uniqueness): Suppose that (q, r) and (q, r ) both satisfy the conclusion of the Theorem. Then kq + r = kq + r 7 since both sides are equal to j. Rearranging gives: (q q ) = r r k. The right-hand side must be an integer (why?) We know that 0 r < k and k < r 0. Adding these together gives k < r r < k. Thus 1 < r r k < 1. What does this tell us?

8 8 (Uniqueness, continued): The only integer between 1 and 1 is zero. We thus have r r k = 0, which immediately gives r = r. This trick is available precisely because we ve restricted ourselves to integer values. It also follow that q q = 0. In other words, q = q. This proof contains a common method for showing that a certain object is unique (in this case, the pair (q, r)): assume you have two things satisfying a given condition, and show that those two things must be equal.

9 Divisibility (Section 2-2) Definition. Given an integer n and a nonzero integer d, we say that d divides n if there is an integer c such that n = cd. We write d n if d divides n, and d n if d does not divide n. 9 Note that d n is equivalent to n d is an integer. However, you are advised to avoid using fractions in problems about divisibility. In terms of the Division Lemma, d n is equivalent to n leaves a remainder of zero when divided by d. Keep this in mind: many divisibility proofs use this idea. Divisibility is one of the key concepts of this course. It leads to the idea of congruence (modulo n), which will be another central theme.

10 Exercise: Properties of Divisibility 10 Prove these directly from the definition (do not use fractions!): If a is a nonzero integer, then a 0 and 1 a. If a b and b c, then a c. [IMPORTANT!] If d a and d b, then d (am + bn) for any m, n Z. If a b and b a, then a = ±b. Find positive integers a, b, c such that a bc, but a b and a c.

11 Greatest Common Divisor/Factor 11 You know from basic arithmetic that, given two integers, we can find their greatest common divisor (or greatest common factor). One way to find a g.c.d. is using prime factorization. For example: 1512 = = gcd(1512, 4410) = = 126 This is not always so simple. Try finding gcd(18, 42) (easy!) gcd(629, 1517) (more work!)

12 We use the following definition of gcd(a, b): Definition (2-1). Let a and b be two integers, not both zero. The greatest common divisor of a and b is a positive integer d which satisfies the following properties: 1. d a and d b. In other words, d is a common divisor of a and b. 2. For any positive integer c for which c a and c b, we have c d. Thus, any common divisor of a and b must also divide d. 12 This definition is easier to use in practice. It can be shown that if d satisfies this definition and c is any common divisor of a and b, then d c.

13 The Euclidean Algorithm 13 It turns out that we can compute a gcd without using prime factorization. For example, we find gcd(1517, 629) by repeated use of the Division Lemma: 1517 =(629)(2) =(259)(2) =(111)(2) =(37)(3) + 0 To show that 37 is a common divisor, start with the last equation and work up. To show any common divisor divides 37, start with the first equation and work down.

14 The Euclidean Algorthim 14 The same process can be used to find gcd(a, b) if a and b are positive: 1. Let DIVIDEND = larger of a and b, DIVISOR = smaller of a and b. 2. Find the REMAINDER of DIVIDEND DIVISOR. 3. If REMAINDER = 0, then DIVISOR is the gcd. [Stop] 4. Otherwise, let DIVIDEND = DIVISOR, then DIVISOR = REMAINDER. 5. Go back to step 2.

15 Exercises: Euclidean Algorithm 15 Use the Euclidean Algorithm to find the following gcd s. Keep all of your work; we will use it for a different purpose later. gcd(90, 25) gcd(2499, 182) gcd(629, 518) gcd(4721, 1361) gcd(89, 55)

16 Theorem (2-2). If a and b are integers, not both zero, then gcd(a, b) exists and is unique. Note that this result is trivial if either a or b is zero. Also note that changing the sign of a or b or swapping a for b does not change the gcd. Without loss of generality, we may thus assume that 0 < b a. For existence, we need to show that the Euclidean Algorithm eventually produces a remainder of zero, at which point the algorithm terminates. This follows from the Division Lemma, noting that the remainder at any stage is always smaller than the divisor at that stage. Since the remainder at one stage becomes the divisor in the next stage, the sequence of remainders decreases by at least 1 at each step. Thus, we have 0 r i b i. So we get a remainder of zero after at most a steps (and usually, much fewer). 16

17 The equation ax + by = c 17 Choose nonzero integers a, b, c. above equation? Are there integers x, y that satisfy the Suppose such integers x and y do exist. Let d = gcd(a, b). Since x and y are integers, the left-hand side is divisible by d. So a necessary condition is that gcd(a, b) c. If this is not true, then we cannot find integers x, y that satisfy the equation. It turns out that this condition is also sufficient. The difficult step is using the Euclidean Algorithm to solve ax + by = d; the rest is very easy.

18 Here is most of our work from gcd(1517, 629) = 37. I ve rearranged each equation on the right to help with the upcoming process: 1517 = (629)(2) = [629] 629 = (259)(2) = 629 2[259] 259 = (111)(2) = 259 2(111) Start with the last equation, substitute the value of 111 and simplify: 37 = 259 2(111) = 259 2(629 2[259]) = 5(259) 2(629). We can now use the same process to eliminate 259: 37 = 5(259) 2(629) = 5(1517 2[629]) 2(629) = 5(1517) 12(629). 18

19 The general case: ax + by = c 19 To find a solution, assuming c = m gcd(a, b): Find x, y such that ax + by = gcd(a, b). You can use the method based on the Euclidean Algorithm, although you might find a solution by lucky guess. Multiply both sides of the equation by m; rearrange to get a(xm) + b(xm) = c. Note: In each case, these methods produce only one solution. We ll see that if there is at least one solution, then there are infinitely many.

11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic

11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic 11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic Bezout s Lemma Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we

More information

ALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers

ALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some

More information

Math 131 notes. Jason Riedy. 6 October, Linear Diophantine equations : Likely delayed 6

Math 131 notes. Jason Riedy. 6 October, Linear Diophantine equations : Likely delayed 6 Math 131 notes Jason Riedy 6 October, 2008 Contents 1 Modular arithmetic 2 2 Divisibility rules 3 3 Greatest common divisor 4 4 Least common multiple 4 5 Euclidean GCD algorithm 5 6 Linear Diophantine

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

The Euclidean Algorithm and Multiplicative Inverses

The Euclidean Algorithm and Multiplicative Inverses 1 The Euclidean Algorithm and Multiplicative Inverses Lecture notes for Access 2009 The Euclidean Algorithm is a set of instructions for finding the greatest common divisor of any two positive integers.

More information

Chapter 5: The Integers

Chapter 5: The Integers c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition

More information

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z: NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,

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

Cool Results on Primes

Cool Results on Primes Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why

More information

Modular Arithmetic Instructor: Marizza Bailey Name:

Modular Arithmetic Instructor: Marizza Bailey Name: Modular Arithmetic Instructor: Marizza Bailey Name: 1. Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find

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

Homework #2 solutions Due: June 15, 2012

Homework #2 solutions Due: June 15, 2012 All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is

More information

COMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635

COMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635 COMP239: Mathematics for Computer Science II Prof. Chadi Assi assi@ciise.concordia.ca EV7.635 The Euclidean Algorithm The Euclidean Algorithm Finding the GCD of two numbers using prime factorization is

More information

Math 109 HW 9 Solutions

Math 109 HW 9 Solutions Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we

More information

2 Elementary number theory

2 Elementary number theory 2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},

More information

Divisibility = 16, = 9, = 2, = 5. (Negative!)

Divisibility = 16, = 9, = 2, = 5. (Negative!) Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties

More information

18 Divisibility. and 0 r < d. Lemma Let n,d Z with d 0. If n = qd+r = q d+r with 0 r,r < d, then q = q and r = r.

18 Divisibility. and 0 r < d. Lemma Let n,d Z with d 0. If n = qd+r = q d+r with 0 r,r < d, then q = q and r = r. 118 18. DIVISIBILITY 18 Divisibility Chapter V Theory of the Integers One of the oldest surviving mathematical texts is Euclid s Elements, a collection of 13 books. This book, dating back to several hundred

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

NOTES ON INTEGERS. 1. Integers

NOTES ON INTEGERS. 1. Integers NOTES ON INTEGERS STEVEN DALE CUTKOSKY The integers 1. Integers Z = {, 3, 2, 1, 0, 1, 2, 3, } have addition and multiplication which satisfy familar rules. They are ordered (m < n if m is less than n).

More information

4 Powers of an Element; Cyclic Groups

4 Powers of an Element; Cyclic Groups 4 Powers of an Element; Cyclic Groups Notation When considering an abstract group (G, ), we will often simplify notation as follows x y will be expressed as xy (x y) z will be expressed as xyz x (y z)

More information

1. multiplication is commutative and associative;

1. multiplication is commutative and associative; Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.

More information

2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?

2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer? Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative

More information

MATH10040 Chapter 1: Integers and divisibility

MATH10040 Chapter 1: Integers and divisibility MATH10040 Chapter 1: Integers and divisibility Recall the basic definition: 1. Divisibilty Definition 1.1. If a, b Z, we say that b divides a, or that a is a multiple of b and we write b a if there is

More information

Mathematical Reasoning & Proofs

Mathematical Reasoning & Proofs Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0

More information

5: The Integers (An introduction to Number Theory)

5: The Integers (An introduction to Number Theory) c Oksana Shatalov, Spring 2017 1 5: The Integers (An introduction to Number Theory) The Well Ordering Principle: Every nonempty subset on Z + has a smallest element; that is, if S is a nonempty subset

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

Number Theory Math 420 Silverman Exam #1 February 27, 2018

Number Theory Math 420 Silverman Exam #1 February 27, 2018 Name: Number Theory Math 420 Silverman Exam #1 February 27, 2018 INSTRUCTIONS Read Carefully Time: 50 minutes There are 5 problems. Write your name neatly at the top of this page. Write your final answer

More information

Writing Assignment 2 Student Sample Questions

Writing Assignment 2 Student Sample Questions Writing Assignment 2 Student Sample Questions 1. Let P and Q be statements. Then the statement (P = Q) ( P Q) is a tautology. 2. The statement If the sun rises from the west, then I ll get out of the bed.

More information

Intermediate Math Circles February 29, 2012 Linear Diophantine Equations I

Intermediate Math Circles February 29, 2012 Linear Diophantine Equations I Intermediate Math Circles February 29, 2012 Linear Diophantine Equations I Diophantine equations are equations intended to be solved in the integers. We re going to focus on Linear Diophantine Equations.

More information

Solution Sheet (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = (i) gcd (97, 157) = 1 = ,

Solution Sheet (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = (i) gcd (97, 157) = 1 = , Solution Sheet 2 1. (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = 3. 2. (i) gcd (97, 157) = 1 = 34 97 21 157, (ii) gcd (527, 697) = 17 = 4 527 3 697, (iii) gcd (2323, 1679) =

More information

2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?

2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer? Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative

More information

Q 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?

Q 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today? 2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a

More information

Contribution of Problems

Contribution of Problems Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

More information

Commutative Rings and Fields

Commutative Rings and Fields Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two

More information

EUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972)

EUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972) Intro to Math Reasoning Grinshpan EUCLID S ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC after N. Vasiliev and V. Gutenmacher (Kvant, 1972) We all know that every composite natural number is a product

More information

The following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers:

The following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers: Divisibility Euclid s algorithm The following is an informal description of Euclid s algorithm for finding the greatest common divisor of a pair of numbers: Divide the smaller number into the larger, and

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

Number Theory Proof Portfolio

Number Theory Proof Portfolio Number Theory Proof Portfolio Jordan Rock May 12, 2015 This portfolio is a collection of Number Theory proofs and problems done by Jordan Rock in the Spring of 2014. The problems are organized first by

More information

Chapter 3 Basic Number Theory

Chapter 3 Basic Number Theory Chapter 3 Basic Number Theory What is Number Theory? Well... What is Number Theory? Well... Number Theory The study of the natural numbers (Z + ), especially the relationship between different sorts of

More information

Number Theory Notes Spring 2011

Number Theory Notes Spring 2011 PRELIMINARIES The counting numbers or natural numbers are 1, 2, 3, 4, 5, 6.... The whole numbers are the counting numbers with zero 0, 1, 2, 3, 4, 5, 6.... The integers are the counting numbers and zero

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

1 Overview and revision

1 Overview and revision MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction

More information

a the relation arb is defined if and only if = 2 k, k

a the relation arb is defined if and only if = 2 k, k DISCRETE MATHEMATICS Past Paper Questions in Number Theory 1. Prove that 3k + 2 and 5k + 3, k are relatively prime. (Total 6 marks) 2. (a) Given that the integers m and n are such that 3 (m 2 + n 2 ),

More information

Lecture Notes. Advanced Discrete Structures COT S

Lecture Notes. Advanced Discrete Structures COT S Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section

More information

WORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:

WORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers: WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their

More information

Basic elements of number theory

Basic elements of number theory Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a

More information

Basic elements of number theory

Basic elements of number theory Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation

More information

CHAPTER 3. Congruences. Congruence: definitions and properties

CHAPTER 3. Congruences. Congruence: definitions and properties CHAPTER 3 Congruences Part V of PJE Congruence: definitions and properties Definition. (PJE definition 19.1.1) Let m > 0 be an integer. Integers a and b are congruent modulo m if m divides a b. We write

More information

PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.

PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime. PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice

More information

2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.

2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. 2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say

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

Rings and modular arithmetic

Rings and modular arithmetic Chapter 8 Rings and modular arithmetic So far, we have been working with just one operation at a time. But standard number systems, such as Z, have two operations + and which interact. It is useful to

More information

Summary: Divisibility and Factorization

Summary: Divisibility and Factorization Summary: Divisibility and Factorization One of the main subjects considered in this chapter is divisibility of integers, and in particular the definition of the greatest common divisor Recall that we have

More information

CS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II

CS 5319 Advanced Discrete Structure. Lecture 9: Introduction to Number Theory II CS 5319 Advanced Discrete Structure Lecture 9: Introduction to Number Theory II Divisibility Outline Greatest Common Divisor Fundamental Theorem of Arithmetic Modular Arithmetic Euler Phi Function RSA

More information

Chapter 5. Number Theory. 5.1 Base b representations

Chapter 5. Number Theory. 5.1 Base b representations Chapter 5 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,

More information

Direct Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24

Direct Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24 Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using

More information

#26: Number Theory, Part I: Divisibility

#26: Number Theory, Part I: Divisibility #26: Number Theory, Part I: Divisibility and Primality April 25, 2009 This week, we will spend some time studying the basics of number theory, which is essentially the study of the natural numbers (0,

More information

Contribution of Problems

Contribution of Problems Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

More information

Integers and Division

Integers and Division Integers and Division Notations Z: set of integers N : set of natural numbers R: set of real numbers Z + : set of positive integers Some elements of number theory are needed in: Data structures, Random

More information

4 PRIMITIVE ROOTS Order and Primitive Roots The Index Existence of primitive roots for prime modulus...

4 PRIMITIVE ROOTS Order and Primitive Roots The Index Existence of primitive roots for prime modulus... PREFACE These notes have been prepared by Dr Mike Canfell (with minor changes and extensions by Dr Gerd Schmalz) for use by the external students in the unit PMTH 338 Number Theory. This booklet covers

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

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

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

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

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

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

Beautiful Mathematics

Beautiful Mathematics Beautiful Mathematics 1. Principle of Mathematical Induction The set of natural numbers is the set of positive integers {1, 2, 3,... } and is denoted by N. The Principle of Mathematical Induction is a

More information

Chapter 14: Divisibility and factorization

Chapter 14: Divisibility and factorization Chapter 14: Divisibility and factorization Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter

More information

Clock Arithmetic and Euclid s Algorithm

Clock Arithmetic and Euclid s Algorithm Clock Arithmetic and Euclid s Algorithm Lecture notes for Access 2008 by Erin Chamberlain. Earlier we discussed Caesar Shifts and other substitution ciphers, and we saw how easy it was to break these ciphers

More information

32 Divisibility Theory in Integral Domains

32 Divisibility Theory in Integral Domains 3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible

More information

Intermediate Math Circles February 26, 2014 Diophantine Equations I

Intermediate Math Circles February 26, 2014 Diophantine Equations I Intermediate Math Circles February 26, 2014 Diophantine Equations I 1. An introduction to Diophantine equations A Diophantine equation is a polynomial equation that is intended to be solved over the integers.

More information

Math 412, Introduction to abstract algebra. Overview of algebra.

Math 412, Introduction to abstract algebra. Overview of algebra. Math 412, Introduction to abstract algebra. Overview of algebra. A study of algebraic objects and functions between them; an algebraic object is typically a set with one or more operations which satisfies

More information

Math Review. for the Quantitative Reasoning measure of the GRE General Test

Math Review. for the Quantitative Reasoning measure of the GRE General Test Math Review for the Quantitative Reasoning measure of the GRE General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important for solving

More information

Numbers, Groups and Cryptography. Gordan Savin

Numbers, Groups and Cryptography. Gordan Savin Numbers, Groups and Cryptography Gordan Savin Contents Chapter 1. Euclidean Algorithm 5 1. Euclidean Algorithm 5 2. Fundamental Theorem of Arithmetic 9 3. Uniqueness of Factorization 14 4. Efficiency

More information

Discrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6

Discrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6 CS 70 Discrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6 1 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes

More information

Math 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition. Todd Cochrane

Math 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition. Todd Cochrane Math 511, Algebraic Systems, Fall 2017 July 20, 2017 Edition Todd Cochrane Department of Mathematics Kansas State University Contents Notation v Chapter 0. Axioms for the set of Integers Z. 1 Chapter 1.

More information

Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010.

Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010. Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010. (24) The p-adic numbers. Let p {2, 3, 5, 7, 11,... } be a prime number. (a) For x Q, define { 0 for x = 0, x p = p n for x = p n (a/b),

More information

CHAPTER 4: EXPLORING Z

CHAPTER 4: EXPLORING Z CHAPTER 4: EXPLORING Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction In this chapter we continue the study of the ring Z. We begin with absolute values. The absolute value function Z N is the identity

More information

Proof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have

Proof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have Exercise 13. Consider positive integers a, b, and c. (a) Suppose gcd(a, b) = 1. (i) Show that if a divides the product bc, then a must divide c. I give two proofs here, to illustrate the different methods.

More information

5 + 9(10) + 3(100) + 0(1000) + 2(10000) =

5 + 9(10) + 3(100) + 0(1000) + 2(10000) = Chapter 5 Analyzing Algorithms So far we have been proving statements about databases, mathematics and arithmetic, or sequences of numbers. Though these types of statements are common in computer science,

More information

MATH 3240Q Introduction to Number Theory Homework 4

MATH 3240Q Introduction to Number Theory Homework 4 If the Sun refused to shine I don t mind I don t mind If the mountains fell in the sea Let it be it ain t me Now if six turned out to be nine Oh I don t mind I don t mind Jimi Hendrix If Six Was Nine from

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

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

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 5

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 5 CS 70 Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 5 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a

More information

Algebra for error control codes

Algebra for error control codes Algebra for error control codes EE 387, Notes 5, Handout #7 EE 387 concentrates on block codes that are linear: Codewords components are linear combinations of message symbols. g 11 g 12 g 1n g 21 g 22

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

Chapter 1. Greatest common divisor. 1.1 The division theorem. In the beginning, there are the natural numbers 0, 1, 2, 3, 4,...,

Chapter 1. Greatest common divisor. 1.1 The division theorem. In the beginning, there are the natural numbers 0, 1, 2, 3, 4,..., Chapter 1 Greatest common divisor 1.1 The division theorem In the beginning, there are the natural numbers 0, 1, 2, 3, 4,..., which constitute the set N. Addition and multiplication are binary operations

More information

GENERATING SETS KEITH CONRAD

GENERATING SETS KEITH CONRAD GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

More information

Some Facts from Number Theory

Some Facts from Number Theory Computer Science 52 Some Facts from Number Theory Fall Semester, 2014 These notes are adapted from a document that was prepared for a different course several years ago. They may be helpful as a summary

More information

Lecture 2. The Euclidean Algorithm and Numbers in Other Bases

Lecture 2. The Euclidean Algorithm and Numbers in Other Bases Lecture 2. The Euclidean Algorithm and Numbers in Other Bases At the end of Lecture 1, we gave formulas for the greatest common divisor GCD (a, b), and the least common multiple LCM (a, b) of two integers

More information

Senior Math Circles Cryptography and Number Theory Week 2

Senior Math Circles Cryptography and Number Theory Week 2 Senior Math Circles Cryptography and Number Theory Week 2 Dale Brydon Feb. 9, 2014 1 Divisibility and Inverses At the end of last time, we saw that not all numbers have inverses mod n, but some do. We

More information

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2 8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose

More information

AN ALGEBRAIC PROOF OF RSA ENCRYPTION AND DECRYPTION

AN ALGEBRAIC PROOF OF RSA ENCRYPTION AND DECRYPTION AN ALGEBRAIC PROOF OF RSA ENCRYPTION AND DECRYPTION Recall that RSA works as follows. A wants B to communicate with A, but without E understanding the transmitted message. To do so: A broadcasts RSA method,

More information

Divisibility. Def: a divides b (denoted a b) if there exists an integer x such that b = ax. If a divides b we say that a is a divisor of b.

Divisibility. Def: a divides b (denoted a b) if there exists an integer x such that b = ax. If a divides b we say that a is a divisor of b. Divisibility Def: a divides b (denoted a b) if there exists an integer x such that b ax. If a divides b we say that a is a divisor of b. Thm: (Properties of Divisibility) 1 a b a bc 2 a b and b c a c 3

More information

AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS

AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS The integers are the set 1. Groups, Rings, and Fields: Basic Examples Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, and we can add, subtract, and multiply

More information

The Chinese Remainder Theorem

The Chinese Remainder Theorem Sacred Heart University DigitalCommons@SHU Academic Festival Apr 20th, 9:30 AM - 10:45 AM The Chinese Remainder Theorem Nancirose Piazza Follow this and additional works at: http://digitalcommons.sacredheart.edu/acadfest

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

1. Algebra 1.5. Polynomial Rings

1. Algebra 1.5. Polynomial Rings 1. ALGEBRA 19 1. Algebra 1.5. Polynomial Rings Lemma 1.5.1 Let R and S be rings with identity element. If R > 1 and S > 1, then R S contains zero divisors. Proof. The two elements (1, 0) and (0, 1) are

More information

Properties of the Integers

Properties of the Integers Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called

More information

a + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c.

a + b = b + a and a b = b a. (a + b) + c = a + (b + c) and (a b) c = a (b c). a (b + c) = a b + a c and (a + b) c = a c + b c. Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called

More information