Homework #2 solutions Due: June 15, 2012

Size: px
Start display at page:

Download "Homework #2 solutions Due: June 15, 2012"

Transcription

1 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 write d = 475s + 385t for some integers s and t. Solution. The greatest common divisor d is computed by applying the Euclidean algorithm: 475 = = = = = = Hence d = gcd(475, 385) = 5. To write d = 5 as a linear combination, d = 475s + 385t, reverse the above Euclidean algorithm computation: 5 = Thus 5 = Alternate Solution: = 15 (25 15) = = 2( ) 25 = = ( ) = = 30( ) = Use the matrix method illustrated on Page 11. The notation k between two matrices means that the second matrix is obtained from the first by subtracting k times the second row from the first, while, if the k is on top (as in k ), then k times the first row is subtracted from the second. [ ] Hence d = (475, 385) = 5 = (a) Calculate d = gcd(4307, 1121) and express it as a linear combination of 4307 and Math

2 Solution. The simplest method is to use the Euclidean algorithm. The following is the matrix format of this algorithm: [ ] Hence d = (4307, 1121) = 59 = (b) Calculate m = lcm[4307, 1121]. Hint: You may want to refer to the formula on page 22 of the handout. Solution. By the formula on page 22 of the handout, we have (a, b)[a, b] = ab for any positive integers a and b. Hence, using part (a): [4307, 1121] = (4307, 1121) = = Which integers can be expressed in the form 12m + 20n, where m and n are integers? Solution. We proved in class that if a and b are natural numbers with at least one not zero, then {ax + by : x, y Z} = az + bz = dz, where d = (a, b). This is essentially the same argument that is used to prove Theorem and Theorem in the handout. Thus, since (12, 20) = 4 it follows that {12m + 20n : m, n Z} = (12, 20)Z = 4Z. Thus the integers that can be expressed in the form 12m + 20n are all of the multiples of Give a proof by induction to show that 5 2n 1 is divisible by 24, for all positive integers n. Solution. Let P (n) be the statement 5 2n 1 is divisible by 24. We will prove that P (n) is true for all positive integers n by induction. The statement P (1) is the statement is divisible by 24. Since = 24, this is certainly a true statement. For the induction step, assume that the statement P (k) is true. That is, assume that 5 2k 1 is divisible by 24. Using this, we must show that it follows that P (k + 1) is also Math

3 true. That is, we must show that 5 2(k+1) 1 is divisible by 24, provided that 5 2k 1 is divisible by 24. Assuming that 5 2k 1 = 24m for some m P, we conclude that 5 (2(k+1) 1 = 5 2k+2 1 = 5 2k = 5 2k 5 2 (5 2 24) (since 1 = ) = 5 2 (5 2k 1) + 24 = 5 2 (24m) + 24 (by the induction hypothesis) = 24(5 2 m + 1). Hence 24 divides 5 2(k+1) 1 provided 24 divides 5 2k 1. This is the statement that P (k + 1) is true, provided P (k) is true, and by the principle of induction, we conclude that P (n) is a true statement for all n P. 5. Determine if each of the following statements is true or false. (a) mod 9 Solution. True since = 3 9. (b) 29 1 mod 7 Solution. False since 29 1 = 30 = ( 5) 7+5, so 29 1 is not divisible by 7. (c) 29 6 mod 7 Solution. True since 29 6 = 35 = ( 5) 7. (d) mod 11 Solution. True since 132 = Find all the solutions (when there are any) of the following linear congruences: (a) 8x 6 mod 14 Solution. gcd(8, 14) = 2 so there are 2 distinct solutions modulo 14. Divide the congruence by 2 to get 4x 3 mod 7. Since [4] 1 7 = [2] 7, it follows that x 6 mod 7. Hence, x 6 mod 14 or x 13 mod 14. (b) 66x 100 mod 121 Solution. Since gcd(66, 121) = 11 and , there are no solutions to this congruence. (c) 21x 14 mod 91 Math

4 Solution. gcd(21, 91) = 7 and 7 14 so there are 91/7 = 13 distinct solutions mod 91. Dividing by 7 gives an equivalent congruence 3x 2 mod 13, which has the solution x 5 mod 13. Thus, the solutions mod 91 are x r mod 91, where r {5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89}. (d) 21x 14 mod 89 Solution. Since gcd(21, 89) = 1 there is a unique solution mod 89. Applying the Euclidean algorithm gives = 1, so that the inverse of 21 modulo 89 is 17. Multiplying the congruence by 17 gives x 17 21x mod Find the inverse of 13 modulo 35 and use it to solve the equation [13] 35 x = [9] 35 in Z 35. Solution. From the Euclidean algorithm, = 1. Hence, [13] 1 35 = [ 8] 35 = [27] 35. Multiplying the equation [13] 35 x = [9] 35 by [13] 1 35 = [ 8] 35 gives x = [ 8] 35 [9] 35 = [ 72] 35 = [ ] 35 = [33] Find the multiplicative inverses of the given elements (if possible). (a) [91] 2565 in Z 2565 Solution. Use the Euclidean algorithm to write = 1. Hence, [91] = [451] (b) [423] 6087 in Z 6087 Solution. Use the Euclidean algorithm to check that (423, 6087) = 3 = 1 so there is not multiplicative inverse of 423 modulo Solve the following system of linear congruences: x 4 (mod 24) x 7 (mod 11) Solution. By inspection or using the Euclidean algorithm, find the equation ( 5) = 1. Then the solutions to the simultaneous congruence are given by x 4 (11 11) + 7(( 5)24) = 356 (mod 264). Math

5 10. In Z 18 find all units and all zero divisors. Solution. The units consist of all [a] 18 such that (a, 18) = 1, while the zero divisors consist of all [a] 18 such that (a, 18) > 1. Thus the units are and the zero divisors are 11. Find {[1] 18, [5] 18, [7] 18, [11] 18, [13] 18, [17] 18 } {[0] 18, [2] 18, [3] 18, [4] 18, [6] 18, [8] 18, [9] 18, [10] 18, [12] 18, [14] 18, [15] 18, [16] 18 }. (a) 6 76 mod 13, Solution mod 13 by Euler s theorem, so write 76 = to get 6 76 = = (6 12 ) (6 2 ) mod 13. (b) mod 11, Solution. By Euler, mod 11. Since 1001 = it follows that = 7 mod 11. (c) 2 25 mod 21, Solution. By Euler, φ(21) = 8 so mod 21. Since 25 = , we have = 2 mod 21. (d) 7 66 mod 120. Solution. Since φ(120) = 32, Euler gives mod 120. Since 66 = it follows that = 49 mod Calculate φ(32), φ(33), φ(120), and φ(384). (φ(n) is the Euler φ function evaluated at n.) Solution. 32 = 2 5 so 4φ(32) = = 16; 33 is prime so φ(33) = ; φ(120) = φ(2 3 )φ(3)φ(5) = ( ) 2 4 = 32; 384 = so φ(284) = φ(2 7 )φ(3) = ( )(3 1) = 128. Math

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

12x + 18y = 50. 2x + v = 12. (x, v) = (6 + k, 2k), k Z.

12x + 18y = 50. 2x + v = 12. (x, v) = (6 + k, 2k), k Z. Math 3, Fall 010 Assignment 3 Solutions Exercise 1. Find all the integral solutions of the following linear diophantine equations. Be sure to justify your answers. (i) 3x + y = 7. (ii) 1x + 18y = 50. (iii)

More information

MATH 4400 SOLUTIONS TO SOME EXERCISES. 1. Chapter 1

MATH 4400 SOLUTIONS TO SOME EXERCISES. 1. Chapter 1 MATH 4400 SOLUTIONS TO SOME EXERCISES 1.1.3. If a b and b c show that a c. 1. Chapter 1 Solution: a b means that b = na and b c that c = mb. Substituting b = na gives c = (mn)a, that is, a c. 1.2.1. Find

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

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

Chuck Garner, Ph.D. May 25, 2009 / Georgia ARML Practice

Chuck Garner, Ph.D. May 25, 2009 / Georgia ARML Practice Some Chuck, Ph.D. Department of Mathematics Rockdale Magnet School for Science Technology May 25, 2009 / Georgia ARML Practice Outline 1 2 3 4 Outline 1 2 3 4 Warm-Up Problem Problem Find all positive

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

SOLUTIONS Math 345 Homework 6 10/11/2017. Exercise 23. (a) Solve the following congruences: (i) x (mod 12) Answer. We have

SOLUTIONS Math 345 Homework 6 10/11/2017. Exercise 23. (a) Solve the following congruences: (i) x (mod 12) Answer. We have Exercise 23. (a) Solve the following congruences: (i) x 101 7 (mod 12) Answer. We have φ(12) = #{1, 5, 7, 11}. Since gcd(7, 12) = 1, we must have gcd(x, 12) = 1. So 1 12 x φ(12) = x 4. Therefore 7 12 x

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

Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some. Notation: b Fact: for all, b, c Z:

Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some. Notation: b Fact: for all, b, c Z: Number Theory Basics Z = {..., 2, 1, 0, 1, 2,...} For, b Z, we say that divides b if z = b for some z Z Notation: b Fact: for all, b, c Z:, 1, and 0 0 = 0 b and b c = c b and c = (b + c) b and b = ±b 1

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

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

All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.

All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1

More information

10 Problem 1. The following assertions may be true or false, depending on the choice of the integers a, b 0. a "

10 Problem 1. The following assertions may be true or false, depending on the choice of the integers a, b 0. a Math 4161 Dr. Franz Rothe December 9, 2013 13FALL\4161_fall13f.tex Name: Use the back pages for extra space Final 70 70 Problem 1. The following assertions may be true or false, depending on the choice

More information

1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.

1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1 2 3 style total Math 415 Examination 3 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. The rings

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

Math 312/ AMS 351 (Fall 17) Sample Questions for Final

Math 312/ AMS 351 (Fall 17) Sample Questions for Final Math 312/ AMS 351 (Fall 17) Sample Questions for Final 1. Solve the system of equations 2x 1 mod 3 x 2 mod 7 x 7 mod 8 First note that the inverse of 2 is 2 mod 3. Thus, the first equation becomes (multiply

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

MATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology.

MATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology. MATH 501 Discrete Mathematics Lecture 6: Number theory Prof. Dr. Slim Abdennadher, slim.abdennadher@guc.edu.eg German University Cairo, Department of Media Engineering and Technology 1 Number theory Number

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

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

Notes on Systems of Linear Congruences

Notes on Systems of Linear Congruences MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the

More information

Definition For a set F, a polynomial over F with variable x is of the form

Definition For a set F, a polynomial over F with variable x is of the form *6. Polynomials Definition For a set F, a polynomial over F with variable x is of the form a n x n + a n 1 x n 1 + a n 2 x n 2 +... + a 1 x + a 0, where a n, a n 1,..., a 1, a 0 F. The a i, 0 i n are the

More information

Part V. Chapter 19. Congruence of integers

Part V. Chapter 19. Congruence of integers Part V. Chapter 19. Congruence of integers Congruence modulo m Let m be a positive integer. Definition. Integers a and b are congruent modulo m if and only if a b is divisible by m. For example, 1. 277

More information

Math Homework # 4

Math Homework # 4 Math 446 - Homework # 4 1. Are the following statements true or false? (a) 3 5(mod 2) Solution: 3 5 = 2 = 2 ( 1) is divisible by 2. Hence 2 5(mod 2). (b) 11 5(mod 5) Solution: 11 ( 5) = 16 is NOT divisible

More information

INTEGERS. In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes.

INTEGERS. In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes. INTEGERS PETER MAYR (MATH 2001, CU BOULDER) In this section we aim to show the following: Goal. Every natural number can be written uniquely as a product of primes. 1. Divisibility Definition. Let a, b

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

MATH 145 Algebra, Solutions to Assignment 4

MATH 145 Algebra, Solutions to Assignment 4 MATH 145 Algebra, Solutions to Assignment 4 1: a) Find the inverse of 178 in Z 365. Solution: We find s and t so that 178s + 365t = 1, and then 178 1 = s. The Euclidean Algorithm gives 365 = 178 + 9 178

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

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

Questionnaire for CSET Mathematics subset 1

Questionnaire for CSET Mathematics subset 1 Questionnaire for CSET Mathematics subset 1 Below is a preliminary questionnaire aimed at finding out your current readiness for the CSET Math subset 1 exam. This will serve as a baseline indicator for

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

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

The Chinese Remainder Theorem

The Chinese Remainder Theorem Sacred Heart University piazzan@mail.sacredheart.edu March 29, 2018 Divisibility Divisibility We say a divides b, denoted as a b, if there exists k Z such that ak = b. Example: Consider 2 6. Then k = 3

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

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

4. Congruence Classes

4. Congruence Classes 4 Congruence Classes Definition (p21) The congruence class mod m of a Z is Example With m = 3 we have Theorem For a b Z Proof p22 = {b Z : b a mod m} [0] 3 = { 6 3 0 3 6 } [1] 3 = { 2 1 4 7 } [2] 3 = {

More information

Mathematics of Cryptography Part I

Mathematics of Cryptography Part I CHAPTER 2 Mathematics of Crptograph Part I (Solution to Practice Set) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinit to positive infinit. The set of

More information

Number Theory. CSS322: Security and Cryptography. Sirindhorn International Institute of Technology Thammasat University CSS322. Number Theory.

Number Theory. CSS322: Security and Cryptography. Sirindhorn International Institute of Technology Thammasat University CSS322. Number Theory. CSS322: Security and Cryptography Sirindhorn International Institute of Technology Thammasat University Prepared by Steven Gordon on 29 December 2011 CSS322Y11S2L06, Steve/Courses/2011/S2/CSS322/Lectures/number.tex,

More information

Signature: (In Ink) UNIVERSITY OF MANITOBA TEST 1 SOLUTIONS COURSE: MATH 2170 DATE & TIME: February 11, 2019, 16:30 17:15

Signature: (In Ink) UNIVERSITY OF MANITOBA TEST 1 SOLUTIONS COURSE: MATH 2170 DATE & TIME: February 11, 2019, 16:30 17:15 PAGE: 1 of 7 I understand that cheating is a serious offence: Signature: (In Ink) PAGE: 2 of 7 1. Let a, b, m, be integers, m > 1. [1] (a) Define a b. Solution: a b iff for some d, ad = b. [1] (b) Define

More information

MATH 420 FINAL EXAM J. Beachy, 5/7/97

MATH 420 FINAL EXAM J. Beachy, 5/7/97 MATH 420 FINAL EXAM J. Beachy, 5/7/97 1. (a) For positive integers a and b, define gcd(a, b). (b) Compute gcd(1776, 1492). (c) Show that if a, b, c are positive integers, then gcd(a, bc) = 1 if and only

More information

The number of ways to choose r elements (without replacement) from an n-element set is. = r r!(n r)!.

The number of ways to choose r elements (without replacement) from an n-element set is. = r r!(n r)!. The first exam will be on Friday, September 23, 2011. The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class

More information

Elementary Properties of the Integers

Elementary Properties of the Integers 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

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

MTH 346: The Chinese Remainder Theorem

MTH 346: The Chinese Remainder Theorem MTH 346: The Chinese Remainder Theorem March 3, 2014 1 Introduction In this lab we are studying the Chinese Remainder Theorem. We are going to study how to solve two congruences, find what conditions are

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

Discrete Mathematics with Applications MATH236

Discrete Mathematics with Applications MATH236 Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet

More information

Name: Solutions Final Exam

Name: Solutions Final Exam Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of

More information

Math 324, Fall 2011 Assignment 7 Solutions. 1 (ab) γ = a γ b γ mod n.

Math 324, Fall 2011 Assignment 7 Solutions. 1 (ab) γ = a γ b γ mod n. Math 324, Fall 2011 Assignment 7 Solutions Exercise 1. (a) Suppose a and b are both relatively prime to the positive integer n. If gcd(ord n a, ord n b) = 1, show ord n (ab) = ord n a ord n b. (b) Let

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

ECE596C: Handout #11

ECE596C: Handout #11 ECE596C: Handout #11 Public Key Cryptosystems Electrical and Computer Engineering, University of Arizona, Loukas Lazos Abstract In this lecture we introduce necessary mathematical background for studying

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

Fall 2017 Test II review problems

Fall 2017 Test II review problems Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and

More information

3.2 Solving linear congruences. v3

3.2 Solving linear congruences. v3 3.2 Solving linear congruences. v3 Solving equations of the form ax b (mod m), where x is an unknown integer. Example (i) Find an integer x for which 56x 1 mod 93. Solution We have already solved this

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

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

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

MATH FINAL EXAM REVIEW HINTS

MATH FINAL EXAM REVIEW HINTS MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any

More information

then the hard copy will not be correct whenever your instructor modifies the assignments.

then the hard copy will not be correct whenever your instructor modifies the assignments. Assignments for Math 2030 then the hard copy will not be correct whenever your instructor modifies the assignments. exams, but working through the problems is a good way to prepare for the exams. It is

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

Objective Type Questions

Objective Type Questions DISTANCE EDUCATION, UNIVERSITY OF CALICUT NUMBER THEORY AND LINEARALGEBRA Objective Type Questions Shyama M.P. Assistant Professor Department of Mathematics Malabar Christian College, Calicut 7/3/2014

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

Mathematics 228(Q1), Assignment 3 Solutions

Mathematics 228(Q1), Assignment 3 Solutions Mathematics 228(Q1), Assignment 3 Solutions Exercise 1.(10 marks)(a) If m is an odd integer, show m 2 1 mod 8. (b) Let m be an odd integer. Show m 2n 1 mod 2 n+2 for all positive natural numbers n. Solution.(a)

More information

Math 319 Problem Set #2 Solution 14 February 2002

Math 319 Problem Set #2 Solution 14 February 2002 Math 39 Problem Set # Solution 4 February 00. (.3, problem 8) Let n be a positive integer, and let r be the integer obtained by removing the last digit from n and then subtracting two times the digit ust

More information

M381 Number Theory 2004 Page 1

M381 Number Theory 2004 Page 1 M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +

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

Coding Theory ( Mathematical Background I)

Coding Theory ( Mathematical Background I) N.L.Manev, Lectures on Coding Theory (Maths I) p. 1/18 Coding Theory ( Mathematical Background I) Lector: Nikolai L. Manev Institute of Mathematics and Informatics, Sofia, Bulgaria N.L.Manev, Lectures

More information

Number theory (Chapter 4)

Number theory (Chapter 4) EECS 203 Spring 2016 Lecture 10 Page 1 of 8 Number theory (Chapter 4) Review Questions: 1. Does 5 1? Does 1 5? 2. Does (129+63) mod 10 = (129 mod 10)+(63 mod 10)? 3. Does (129+63) mod 10 = ((129 mod 10)+(63

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

The Chinese Remainder Theorem

The Chinese Remainder Theorem The Chinese Remainder Theorem R. C. Daileda February 19, 2018 1 The Chinese Remainder Theorem We begin with an example. Example 1. Consider the system of simultaneous congruences x 3 (mod 5), x 2 (mod

More information

Number Theory. Modular Arithmetic

Number Theory. Modular Arithmetic Number Theory The branch of mathematics that is important in IT security especially in cryptography. Deals only in integer numbers and the process can be done in a very fast manner. Modular Arithmetic

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

CS280, Spring 2004: Prelim Solutions

CS280, Spring 2004: Prelim Solutions CS280, Spring 2004: Prelim Solutions 1. [3 points] What is the transitive closure of the relation {(1, 2), (2, 3), (3, 1), (3, 4)}? Solution: It is {(1, 2), (2, 3), (3, 1), (3, 4), (1, 1), (2, 2), (3,

More information

0 Sets and Induction. Sets

0 Sets and Induction. Sets 0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set

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 261 Spring 2014 Final Exam May 5, 2014

Math 261 Spring 2014 Final Exam May 5, 2014 Math 261 Spring 2014 Final Exam May 5, 2014 1. Give a statement or the definition for ONE of the following in each category. Circle the letter next to the one you want graded. For an extra good final impression,

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

MATH 3240Q Introduction to Number Theory Homework 5

MATH 3240Q Introduction to Number Theory Homework 5 The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and

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

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

SOLUTIONS TO PROBLEM SET 1. Section = 2 3, 1. n n + 1. k(k + 1) k=1 k(k + 1) + 1 (n + 1)(n + 2) n + 2,

SOLUTIONS TO PROBLEM SET 1. Section = 2 3, 1. n n + 1. k(k + 1) k=1 k(k + 1) + 1 (n + 1)(n + 2) n + 2, SOLUTIONS TO PROBLEM SET 1 Section 1.3 Exercise 4. We see that 1 1 2 = 1 2, 1 1 2 + 1 2 3 = 2 3, 1 1 2 + 1 2 3 + 1 3 4 = 3 4, and is reasonable to conjecture n k=1 We will prove this formula by induction.

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

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

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

Mathematics for Computer Science Exercises for Week 10

Mathematics for Computer Science Exercises for Week 10 Mathematics for Computer Science Exercises for Week 10 Silvio Capobianco Last update: 7 November 2018 Problems from Section 9.1 Problem 9.1. Prove that a linear combination of linear combinations of integers

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

7.2 Applications of Euler s and Fermat s Theorem.

7.2 Applications of Euler s and Fermat s Theorem. 7.2 Applications of Euler s and Fermat s Theorem. i) Finding and using inverses. From Fermat s Little Theorem we see that if p is prime and p a then a p 1 1 mod p, or equivalently a p 2 a 1 mod p. This

More information

Number Theory Homework.

Number Theory Homework. Number Theory Homewor. 1. The Theorems of Fermat, Euler, and Wilson. 1.1. Fermat s Theorem. The following is a special case of a result we have seen earlier, but as it will come up several times in this

More information

Numbers. Çetin Kaya Koç Winter / 18

Numbers. Çetin Kaya Koç   Winter / 18 Çetin Kaya Koç http://koclab.cs.ucsb.edu Winter 2016 1 / 18 Number Systems and Sets We represent the set of integers as Z = {..., 3, 2, 1,0,1,2,3,...} We denote the set of positive integers modulo n as

More information

Chapter 4 Finite Fields

Chapter 4 Finite Fields Chapter 4 Finite Fields Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers what constitutes a number

More information

MATH 310: Homework 7

MATH 310: Homework 7 1 MATH 310: Homework 7 Due Thursday, 12/1 in class Reading: Davenport III.1, III.2, III.3, III.4, III.5 1. Show that x is a root of unity modulo m if and only if (x, m 1. (Hint: Use Euler s theorem and

More information

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same

More information

CS2800 Questions selected for fall 2017

CS2800 Questions selected for fall 2017 Discrete Structures Final exam sample questions Solutions CS2800 Questions selected for fall 2017 1. Determine the prime factorizations, greatest common divisor, and least common multiple of the following

More information

CPSC 467b: Cryptography and Computer Security

CPSC 467b: Cryptography and Computer Security CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 8 February 1, 2012 CPSC 467b, Lecture 8 1/42 Number Theory Needed for RSA Z n : The integers mod n Modular arithmetic GCD Relatively

More information

Greatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730

Greatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730 MATH 2730 Greatest Common Divisor Benjamin V.C. Collins James A. Swenson The world s least necessary definition Definition Let a, b Z, not both zero. The largest integer d such that d a and d b is called

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

Chapter 8. Introduction to Number Theory

Chapter 8. Introduction to Number Theory Chapter 8 Introduction to Number Theory CRYPTOGRAPHY AND NETWORK SECURITY 1 Index 1. Prime Numbers 2. Fermat`s and Euler`s Theorems 3. Testing for Primality 4. Discrete Logarithms 2 Prime Numbers 3 Prime

More information

Solutions to Practice Final

Solutions to Practice Final s to Practice Final 1. (a) What is φ(0 100 ) where φ is Euler s φ-function? (b) Find an integer x such that 140x 1 (mod 01). Hint: gcd(140, 01) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100 ) = (

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