# Applications of Fermat s Little Theorem and Congruences

Size: px
Start display at page:

## Transcription

1 Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4 mod 2, 14 0 mod 7, mod 9, mod 35. Properties: Let m be a positive integer and let a, b, c, d be integers. Then 1. a a mod m 2. If a b mod m, then b a mod m. 3. If a b mod m and b c mod m, then a c mod m. 4. (a) If a qm + r mod m, then a r mod m. (b) Every integer a is congruent mod m to exactly one of 0, 1,..., m If a b mod m and c d mod m, then a ± c b ± d mod m and ac bd mod m. 5. If a b mod m, then a ± c b ± c mod m and ac bc mod m. 5. If a b mod m, then a n b n mod m for any n Z If (c, m) = 1 and ac bc mod m, then a b mod m. Theorem (Fermat s Little Theorem): Let p be a prime. Then n p n mod p for any integer n 1. Corollary: Let p be a prime. Then n p 1 1 mod p for any integer n 1 with (n, p) = 1. 1

2 PROBLEMS 1. Find all solutions to each of the following congruences: (i) 2x 1 mod 3. (ii) 3x 4 mod 8. (iii) 6x 3 mod 15. (iv) 8x 7 mod 18. (v) 9x mod What is the last digit of ? 3. Prove that there is no perfect square a 2 which is congruent to 2 mod Prove that there is no perfect square a 2 whose last digit is Prove that is not a perfect square. 6. Prove that there is no perfect square a 2 whose last digits are Prove that the following equations have no solutions in integer numbers: (i) x 2 3y = 5 (ii) 3x 2 4y = 5 (iii) x 2 y 2 = Prove that Prove that Prove that 17 a 80 1 for any a Z + with (a, 17) = What is the remainder after dividing 3 50 by 7? 2

3 THEOREM AND EXAMPLES Theorem: If (a, m) = 1, then, for every integer b, the congruence has exactly one solution where s is such number that ax b mod m (1) x bs mod m, (2) as 1 mod m. (3) Proof (Sketch): We show that (2) is the solution of (1). In fact, if we multiply (2) by a and (3) by b (we can do that by property 5 ), we get which imply (1) by property 3. ax abs mod m and bsa b mod m, Example 1: Find all solutions of the following congruence 2x 5 mod 7. Solution: We first note that (2, 7) = 1. Therefore we can apply the theorem above. Since mod 7, we get x mod 7. Example 2: Find all solutions of the following congruence 2x 5 mod 8. Solution: Since (2, 8) = 2, we can t apply the theorem above directly. We now note that 2x 5 mod 8 is equivalent to 2x 8y = 5, which is impossible, since the left-hand side is divisible by 2, whereas the right-hand side is not. So, this equation has no solutions. Example 3: Find all solutions of the following congruence 4x 2 mod 6. Solution: Since (4, 6) = 2, we can t apply the theorem above directly again. However, canceling out 2 (think about that!), we obtain 2x 1 mod 3. Note that (2, 3) = 1. Therefore we can apply the theorem above to the new equation. Since mod 3, we get x mod 3. Example 4: What is the last digit of ? Solution: It is obvious that mod 10, therefore by property 5 we have This means that the last digit of is mod 10. 3

4 Example 5: Prove that there is no integer number a such that a 4 is congruent to 3 mod 4. Solution: By the property 4(a) each integer number is congruent to 0, 1, 2, or 3 mod 4. Consider all these cases and use property 5 : If a 0 mod 4, then a mod 4. If a 1 mod 4, then a mod 4. If a 2 mod 4, then a mod 4. If a 3 mod 4, then a mod 4. So, a 4 0 or 1 mod 4. Therefore a 4 3 mod 4. Example 6: Prove that there is no perfect square a 2 whose last digit is 3. Solution: By the property 4(a) each integer number is congruent to 0, 1, 2,..., 8 or 9 mod 10. Consider all these cases and use property 5 : If a 0 mod 10, then a mod 10. If a 1 mod 10, then a mod 10. If a 2 mod 10, then a mod 10. If a 3 mod 10, then a mod 10. If a 4 mod 10, then a mod 10. If a 5 mod 10, then a mod 10. If a 6 mod 10, then a mod 10. If a 7 mod 10, then a mod 10. If a 8 mod 10, then a mod 10. If a 9 mod 10, then a mod 10. So, a 2 0, 1, 4, 5, 6 or 9 mod 10. Therefore a 2 3 mod 10, and the result follows. Example 7: Prove that is not a perfect square. Solution: The last digit is 3, which is impossible by Example 6. Example 8: Prove that the equation x 4 4y = 3 has no solutions in integer numbers. Solution: Rewrite this equation as x 4 = 4y + 3, which means that x 4 3 mod 4, which is impossible by Example 5. Example 9: Prove that Solution: We have therefore by property 5 we get which means that mod 10, mod 10, Example 10: Prove that 23 a for any a Z + with (a, 23) = 1. Solution: By Fermat s Little theorem we have a 22 1 mod 23, therefore by property 5 we get and the result follows. a mod 23, 4

5 SOLUTIONS Problem 1(i): Find all solutions of the congruence 2x 1 mod 3. Solution: We first note that (2, 3) = 1. Therefore we can apply the theorem above. Since mod 3, we get x mod 3. Problem 1(ii): Find all solutions of the congruence 3x 4 mod 8. Solution: We first note that (3, 8) = 1. Therefore we can apply the theorem above. Since mod 8, we get x mod 8. Problem 1(iii): Find all solutions of the congruence 6x 3 mod 15. Solution: Since (6, 15) = 3, we can t apply the theorem above directly again. However, canceling out 3, we obtain 2x 1 mod 5. Note that (2, 5) = 1. Therefore we can apply the theorem above to the new equation. Since mod 5, we get x mod 5. Problem 1(iv): Find all solutions of the congruence 8x 7 mod 18. Solution: Since (8, 18) = 2, we can t apply the theorem above directly. We now note that 8x 7 mod 18 is equivalent to 8x 18y = 7, which is impossible, since the left-hand side is divisible by 2, whereas the right-hand side is not. So, this equation has no solutions. Problem 1(v): Find all solutions of the congruence 9x mod 25. Solution: We first rewrite this congruence as 9x 5 mod 25. Note that (9, 25) = 1. Therefore we can apply the theorem above. Since mod 25, we get x mod 25. Problem 2: What is the last digit of ? Solution: It is obvious that mod 10, therefore by property 5 we have mod 10. This means that the last digit is 1. Problem 3: Prove that there is no perfect square a 2 which is congruent to 2 mod 4. Solution: By the property 4(a) each integer number is congruent to 0, 1, 2, or 3 mod 4. Consider all these cases and use property 5 : If a 0 mod 4, then a mod 4. If a 1 mod 4, then a mod 4. If a 2 mod 4, then a mod 4. If a 3 mod 4, then a mod 4. So, a 2 0 or 1 mod 4. Therefore a 2 2 mod 4. Problem 4: Prove that there is no perfect square a 2 whose last digit is 2. Solution: By the property 4(a) each integer number is congruent to 0, 1, 2,..., 8 or 9 mod 10. Consider all these cases and use property 5 : If a 0 mod 10, then a mod 10. If a 1 mod 10, then a mod 10. If a 2 mod 10, then a mod 10. If a 3 mod 10, then a mod 10. If a 4 mod 10, then a mod 10. If a 5 mod 10, then a mod 10. If a 6 mod 10, then a mod 10. If a 7 mod 10, then a mod 10. If a 8 mod 10, then a mod 10. If a 9 mod 10, then a mod 10. So, a 2 0, 1, 4, 5, 6 or 9 mod 10. Therefore a 2 2 mod 10, and the result follows. 5

6 Problem 5: Prove that is not a perfect square. Solution 1: We have = 4k + 2. Therefore it is congruent to 2 mod 4 by property 4(a), which is impossible by Problem 3. Solution 2: The last digit is 2, which is impossible by Problem 4. Problem 6 : Prove that there is no perfect square a 2 whose last digits are 85. Solution: It follows from problem 4 that a 2 5 mod 10 only if a 5 mod 10. Therefore a 2 85 mod 100 only if a 5, 15, 25,..., 95 mod 100. If we consider all these cases and use property 5 is the same manner as in problem 4, we will see that a 2 25 mod 100. Therefore a 2 85 mod 100, and the result follows. Problem 7(i): Prove that the equation x 2 3y = 5 has no solutions in integer numbers. Solution: Rewrite this equation as x 2 = 3y + 5, which means that x mod 3. By the property 4(a) each integer number is congruent to 0, 1, or 2 mod 3. Consider all these cases and use property 5 : If a 0 mod 3, then a mod 3. If a 1 mod 3, then a mod 3. If a 2 mod 4, then a mod 3. So, a 2 0 or 1 mod 3. Therefore a 2 2 mod 3. Problem 7(ii): Prove that the equation 3x 2 4y = 5 has no solutions in integer numbers. Solution: Rewrite this equation as 3x 2 = 4y + 5, which means that 3x mod 4. On the other hand, by Problem 3 we have x 2 0 or 1 mod 4, hence 3x 2 0 or 3 mod 4. Therefore x 2 1 mod 4. Problem 7(iii): Prove that the equation x 2 y 2 = 2002 has no solutions in integer numbers. Solution: By Problem 3 we have x 2 0 or 1 mod 4, hence x 2 y 2 0, 1 or -1 mod 4. On the other hand, mod 4. Therefore x 2 y mod 4, Problem 8: Prove that Solution: We have 11 1 mod 10, therefore by property 5 we get mod 10, which means that Problem 9 : Prove that Solution: We have = ( )(11 5 1) = ( )(11 1)( ). ( ) Since 11 1 mod 10, by property 5 we get 11 n 1 mod 10. Therefore by property 5 we obtain mod 10. Note that is divisible by 2 and 11 1 is divisible by 10. Therefore the right-hand side of ( ) is divisible by = 100. On the other hand, by Fermat s Little Theorem, is divisible by 11. Since (11, 100) = 1, the whole expression is divisible by Problem 10: Prove that 17 a 80 1 for any a Z + with (a, 17) = 1. Solution: By Fermat s Little theorem we have a 16 1 mod 17, therefore by property 5 we get a mod 17, and the result follows. Problem 11: What is the remainder after dividing 3 50 by 7? Solution: By Fermat s Little theorem we have mod 7, therefore by property 5 we get mod 7, therefore mod 7. 6

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples

On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### 3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

### Computing exponents modulo a number: Repeated squaring

Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography

V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 3 Congruence Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical

### GREATEST COMMON DIVISOR

DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory Divisibility and Primes Definition. 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 or divisor of a and write b a. Definition

### Lectures on Number Theory. Lars-Åke Lindahl

Lectures on Number Theory Lars-Åke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

Chapter 2 Remodulization of Congruences Proceedings NCUR VI. è1992è, Vol. II, pp. 1036í1041. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department

### SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

### Discrete Mathematics, Chapter 4: Number Theory and Cryptography

Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### Overview of Number Theory Basics. Divisibility

Overview of Number Theory Basics Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Divisibility Definition Given integers a and b, b 0, b divides a (denoted b a) if integer c, s.t. a = cb. b is called

### Primality - Factorization

Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

### Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms

The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s about

### The Mean Value Theorem

The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

### Number Theory: A Mathemythical Approach. Student Resources. Printed Version

Number Theory: A Mathemythical Approach Student Resources Printed Version ii Contents 1 Appendix 1 2 Hints to Problems 3 Chapter 1 Hints......................................... 3 Chapter 2 Hints.........................................

### calculating the result modulo 3, as follows: p(0) = 0 3 + 0 + 1 = 1 0,

Homework #02, due 1/27/10 = 9.4.1, 9.4.2, 9.4.5, 9.4.6, 9.4.7. Additional problems recommended for study: (9.4.3), 9.4.4, 9.4.9, 9.4.11, 9.4.13, (9.4.14), 9.4.17 9.4.1 Determine whether the following polynomials

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

### Continued Fractions. Darren C. Collins

Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### 4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

### 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

### 3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Some practice problems for midterm 2

Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a long-winded way of asking for the value of 6 2000 mod

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### Online EFFECTIVE AS OF JANUARY 2013

2013 A and C Session Start Dates (A-B Quarter Sequence*) 2013 B and D Session Start Dates (B-A Quarter Sequence*) Quarter 5 2012 1205A&C Begins November 5, 2012 1205A Ends December 9, 2012 Session Break

### ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,

### 1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

### Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS

MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS Class Meetings: MW 2:00-3:15 pm in Physics 144, September 7 to December 14 [Thanksgiving break November 23 27; final exam December 21] Instructor:

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### TEXAS A&M UNIVERSITY. Prime Factorization. A History and Discussion. Jason R. Prince. April 4, 2011

TEXAS A&M UNIVERSITY Prime Factorization A History and Discussion Jason R. Prince April 4, 2011 Introduction In this paper we will discuss prime factorization, in particular we will look at some of the

### 2 When is a 2-Digit Number the Sum of the Squares of its Digits?

When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### Chapter 3. if 2 a i then location: = i. Page 40

Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)

Chapter 4 Complementary Sets Of Systems Of Congruences Proceedings NCUR VII. è1993è, Vol. II, pp. 793í796. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field

Math 345-60 Abstract Algebra I Questions for Section 23: Factoring Polynomials over a Field 1. Throughout this section, F is a field and F [x] is the ring of polynomials with coefficients in F. We will

### ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### RSA and Primality Testing

and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2

### Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

### 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.

### DIVISIBILITY AND GREATEST COMMON DIVISORS

DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1 Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties Then we will

### Handout NUMBER THEORY

Handout of NUMBER THEORY by Kus Prihantoso Krisnawan MATHEMATICS DEPARTMENT FACULTY OF MATHEMATICS AND NATURAL SCIENCES YOGYAKARTA STATE UNIVERSITY 2012 Contents Contents i 1 Some Preliminary Considerations

### Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

### Factoring Polynomials

Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

### Maths delivers! A guide for teachers Years 11 and 12. RSA Encryption

1 Maths delivers! 2 3 4 5 6 7 8 9 10 11 12 A guide for teachers Years 11 and 12 RSA Encryption Maths delivers! RSA Encryption Dr Michael Evans AMSI Editor: Dr Jane Pitkethly, La Trobe University Illustrations

### Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

### 2 : two cube. 5 : five cube. 10 : ten cube.

Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VI Instructor: Line #: 52920 Yasuyuki Kachi 6 Cubes February 2 Mon, 2015 We can similarly define the notion of cubes/cubing Like we did last time, 3 2

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com

Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### FACTORING AFTER DEDEKIND

FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents

### Pythagorean vectors and their companions. Lattice Cubes

Lattice Cubes Richard Parris Richard Parris (rparris@exeter.edu) received his mathematics degrees from Tufts University (B.A.) and Princeton University (Ph.D.). For more than three decades, he has lived

### FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set

FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,

### 3 Factorisation into irreducibles

3 Factorisation into irreducibles Consider the factorisation of a non-zero, non-invertible integer n as a product of primes: n = p 1 p t. If you insist that primes should be positive then, since n could

### COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:

COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative

### Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

### 26 Ideals and Quotient Rings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

### Settling a Question about Pythagorean Triples

Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:

### Lecture Notes on Polynomials

Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

### FACTORING OUT COMMON FACTORS

278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation

CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25