Number Theory and Divisibility


 Maurice Watts
 2 years ago
 Views:
Transcription
1
2 Number Theory and Divisibility Recall the Natural Numbers: N = {1, 2, 3, 4, 5, 6, } Any Natural Number can be expressed as the product of two or more Natural Numbers: 2 x 12 = 24 3 x 8 = 24 6 x 4 = 24 Factors of 24 Factors of 24 Factors of 24 Factors of 24: Natural Numbers that multiply to give us 24
3 Number Theory and Divisibility Factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24 1 x 24 = 24 2 x 12 = 24 3 x 8 = 24 4 x 6 = 24 6 x 4 = 24 8 x 3 = x 2 = x 1 = 24
4 Divisibility a is divisible by b, if the operation of dividing a by b leaves a remainder of 0 b a b is a divisor of a b divides a The following are Equivalent Statements: 24 is divisible by 8 8 is a divisor of 24 8 divides
5 Ex. Divisors of 24 Divisibility 1 24 because 24 1 = 24 with no remainder 2 24 because 24 2 = 12 with no remainder 3 24 because 24 3 = 8 with no remainder 4 24 because 24 4 = 6 with no remainder 6 24 because 24 6 = 4 with no remainder 8 24 because 24 8 = 3 with no remainder because = 2 with no remainder because = 1 with no remainder 5 24 because 24 5 gives us a nonzero remainder 7 24 because 24 5 gives us a nonzero remainder
6 Divisibility True or False? 4 16 True since 16 4 = False since we get a remainder 7 14 True since 14 7 = True since 15 3 = True since = 104
7 Divisibility Rules
8 Divisibility True or False? Use the Rules of Divisibility 8 48,324 False, last 3 digits 324 not divisible by ,324 True, divisible by 2 and ,324 False, 4 does divide 48,324 since 4 24
9 Prime Factorization Prime Number: a Natural Number greater than 1 that has only itself and 1 as factors. Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, Composite Number: a Natural Number greater than 1 that is divisible by a number other than 1 and itself. Ex. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, Note: 1 is neither prime nor composite by definition
10 Prime Factorization Every Composite Number can be expressed as the product of prime numbers. Expressing a number in this form is called Prime Factorization. Ex. 45 = 3 x 3 x 5 18 = 2 x 3 x 3 42 = 7 x 3 x 2
11 Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of prime numbers in one and only one way. (Order does not matter) This means Prime Factorization is unique to every composite number.
12 Factor Tree A method to find the prime factorization of a composite number Step 1) Select two numbers (other than 1) that are factors of your number Step 2) Repeat this process for the factors that are composite numbers until you only have primes left
13 Factor Tree Ex. Find the prime factorization of = 2 2 x 5 2 x 7 Ex. Find the prime factorization of 120
14 Greatest Common Divisor gcd(a,b) = the largest number that is a divisor (factor) of a and b. If gcd(a,b) = 1, then a and b are Relatively Prime Finding GCD using Prime Factorization 1) Write prime factorization of each number 2) Select each prime factor with the smallest exponent that is common to each prime factorizations 3) The GCD is the product of the numbers in part 2
15 Greatest Common Divisor Find the GCD of 16 and = 2 2 x 3 16 = 2 4 gcd(12,16) = 2 2 = 4
16 Greatest Common Divisor Find the GCD of 40 and 24
17 Greatest Common Divisor For an intramural league, you need to divide 192 men and 288 women into allmale and allfemale teams so that each team has the same number of people. What is the largest number of people that can be placed on a team?
18 Greatest Common Divisor 192 men divided into teams: number of men per team is a divisor of women divided into teams: number of women per team is a divisor of 288 The number of people per team must be the same for both men and women. We are looking for the largest number for this occur, i.e. the largest number that divides both 192 and 288 without a remainder
19 Greatest Common Divisor We want to find the GCD of 192 and Therefore, the greatest number of people that can be placed into teams is 2 5 x 3 = 96
20 Least Common Multiple lcm(a,b) = smallest Natural Number that is divisible by a and b Find LCM by making a list of multiples of each number Ex. Find lcm(15,20) Multiples of 15: {15, 30, 45, 60, 75, 90, 105, 120, } Multiples of 20: {20, 40, 60, 80, 100, 120, 140, 160, } 60 and 120 are common multiples. The least common multiple is 60.
21 Least Common Multiple Method 2: Finding LCM using Prime Factorization 1) Write the prime factorization of each number 2) Select every prime raised to the greatest power that it occurs 3) The product of numbers in Step 2 is the LCM
22 Least Common Multiple Find the LCM of 16 and 12 Recall that: 12 = 2 2 x 3 and 16 = 2 4 Prime 2 has highest exponent 4 Prime 3 has highest exponent 1 Therefore lcm(16,12) = 2 4 x 3 = 48
23 Least Common Multiple A movie theater runs two documentary films continuously. One documentary runs for 40 minutes and a second documentary runs for 60 minutes. Both movies begin at 3PM. When will the movies begin again at the same time?
24 Least Common Multiple The documentaries repeat and we want to find when they both start at the same time. Common multiples of 40 and 60 will give us the minutes after 3PM when both movies start at the same time. We want to find the LCM, which is the first time they start together again. lcm(40,60) = minutes = 2 hours 2 hours from 3PM is 5PM
25 Practice Problems Page 256 Divisible? #1, 7, 11, 17, 21 Prime Factorization: #25, 27, 31, 35, 39, 41 GCD: #45, 49, 51, 53, 55 LCM: #57, 59, 61, 65, 67 Applications: #91, 98
N= {1,2,3,4,5,6,7,8,9,10,11,...}
1.1: Integers and Order of Operations 1. Define the integers 2. Graph integers on a number line. 3. Using inequality symbols < and > 4. Find the absolute value of an integer 5. Perform operations with
More informationDiscrete Structures Lecture Primes and Greatest Common Divisor
DEFINITION 1 EXAMPLE 1.1 EXAMPLE 1.2 An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite.
More informationDivisibility, Factors, and Multiples
Divisibility, Factors, and Multiples An Integer is said to have divisibility with another nonzero Integer if it can divide into the number and have a remainder of zero. Remember: Zero divided by any number
More informationCh 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 information2 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 informationThe 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 information5.1. Primes, Composites, and Tests for Divisibility
CHAPTER 5 Number Theory 5.1. Primes, Composites, and Tests for Divisibility Definition. A counting number with exactly two di erent factors is called a prime number or a prime. A counting number with more
More informationArithmetic, Algebra, Number Theory
Arithmetic, Algebra, Number Theory Peter Simon 21 April 2004 Types of Numbers Natural Numbers The counting numbers: 1, 2, 3,... Prime Number A natural number with exactly two factors: itself and 1. Examples:
More informationSlide 1 / 69. Slide 2 / 69. Slide 3 / 69. Whole Numbers. Table of Contents. Prime and Composite Numbers
Slide 1 / 69 Whole Numbers Table of Contents Slide 2 / 69 Prime and Composite Numbers Prime Factorization Common Factors Greatest Common Factor Relatively Prime Least Common Multiple Slide 3 / 69 Prime
More informationSEVENTH EDITION and EXPANDED SEVENTH EDITION
SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 51 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to
More informationIntroduction to Number Theory
Introduction to Number Theory Number theory is about integers and their properties. We will start with the basic principles of divisibility, greatest common divisors, least common multiples, and modular
More informationCISC102 Fall 2017 Week 6
Week 6 page 1! of! 15 CISC102 Fall 2017 Week 6 We will see two different, yet similar, proofs that there are infinitely many prime numbers. One proof would surely suffice. However, seeing two different
More informationa 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 information8 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 informationPrime Factorization and GCF. In my own words
Warm up Problem What is a prime number? A PRIME number is an INTEGER greater than 1 with EXACTLY 2 positive factors, 1 and the number ITSELF. Examples of prime numbers: 2, 3, 5, 7 What is a composite
More informationThe numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory.
6.1 Number Theory Number Theory The numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory. 2010 Pearson Education,
More informationREVIEW Chapter 1 The Real Number System
REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }
More informationDecimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.
Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before
More informationProofs. Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm. Reading (Epp s textbook)
Proofs Methods of Proof Divisibility Floor and Ceiling Contradiction & Contrapositive Euclidean Algorithm Reading (Epp s textbook) 4.3 4.8 1 Divisibility The notation d n is read d divides n. Symbolically,
More informationLP03 Chapter 5. A prime number is a natural number greater that 1 that has only itself and 1 as factors. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
LP03 Chapter 5 Prime Numbers A prime number is a natural number greater that 1 that has only itself and 1 as factors. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, Question 1 Find the prime factorization of 120.
More informationQuantitative Aptitude
WWW.UPSCMANTRA.COM Quantitative Aptitude Concept 1 1. Number System 2. HCF and LCM 2011 Prelims Paper II NUMBER SYSTEM 2 NUMBER SYSTEM In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7,
More informationNumber Theory. Number Theory. 6.1 Number Theory
6.1 Number Theory Number Theory The numbers 1, 2, 3, are called the counting numbers or natural numbers. The study of the properties of counting numbers is called number theory. 2 2010 Pearson Education,
More informationMath 110 FOUNDATIONS OF THE REAL NUMBER SYSTEM FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS
41Divisibility Divisibility Divisibility Rules Divisibility An integer is if it has a remainder of 0 when divided by 2; it is otherwise. We say that 3 divides 18, written, because the remainder is 0 when
More informationMTH 310, Section 001 Abstract Algebra I and Number Theory. Sample Midterm 1
MTH 310, Section 001 Abstract Algebra I and Number Theory Sample Midterm 1 Instructions: You have 50 minutes to complete the exam. There are five problems, worth a total of fifty points. You may not use
More informationNumber Sense. Basic Ideas, Shortcuts and Problems #120 from the Sequence Chart
UIL Number Sense Contest Basic Ideas, Shortcuts and Problems #120 from the Sequence Chart Larry White UIL State Number Sense Contest Director texasmath@centex.net http://www.uiltexas.org/academics/numbersense
More informationNumber 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 informationThe 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 informationFinding Prime Factors
Section 3.2 PREACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to
More informationHomework #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 informationNOTES 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 informationSection 34: Least Common Multiple and Greatest Common Factor
Section : Fraction Terminology Identify the following as proper fractions, improper fractions, or mixed numbers:, proper fraction;,, improper fractions;, mixed number. Write the following in decimal notation:,,.
More informationMath.3336: Discrete Mathematics. Primes and Greatest Common Divisors
Math.3336: Discrete Mathematics Primes and Greatest Common Divisors Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu
More informationDiscrete Math. Instructor: Mike Picollelli. Day 10
Day 10 Fibonacci Redux. Last time, we saw that F n = 1 5 (( 1 + ) n ( 5 2 1 ) n ) 5. 2 What Makes The Fibonacci Numbers So Special? The Fibonacci numbers are a particular type of recurrence relation, a
More informationDiscrete Mathematics GCD, LCM, RSA Algorithm
Discrete Mathematics GCD, LCM, RSA Algorithm Abdul Hameed http://informationtechnology.pk/pucit abdul.hameed@pucit.edu.pk Lecture 16 Greatest Common Divisor 2 Greatest common divisor The greatest common
More informationHCF & LCM Solved Sums
HCF & LCM Solved Sums 1. The least common multiple of 24, 36, and 40 is A ) 340 b ) 360 c) 230 d) 400 2 24 36 40 => 2 X 2 X 2 X 3 X 1 X 3 X 5 2 12 18 20 => 360 2 6 9 10 3 3 9 5 1 3 5 Ans : 360 2. The LCM
More informationMath Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions
Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warmup Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number
More information18 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 informationSection 1.3 Review of Complex Numbers
1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that
More informationChapter 3: The Euclidean Algorithm and Diophantine. Math 138 Burger California State University, Fresno
Chapter 3: The Euclidean Algorithm and Diophantine Equations Math 138 Burger California State University, Fresno Greatest Common Divisor d is the greatest common divisor of integers a and b if d is the
More informationQuiz 1, Mon CS 2050, Intro Discrete Math for Computer Science
Quiz 1, Mon 09611 CS 050, Intro Discrete Math for Computer Science This quiz has 10 pages (including this cover page) and 5 Problems: Problems 1,, 3 and 4 are mandatory ( pages each.) Problem 5 is optional,
More informationKNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS
DOMAIN I. COMPETENCY 1.0 MATHEMATICS KNOWLEDGE OF NUMBER SENSE, CONCEPTS, AND OPERATIONS Skill 1.1 Compare the relative value of real numbers (e.g., integers, fractions, decimals, percents, irrational
More information1. (16 points) Circle T if the corresponding statement is True or F if it is False.
Name Solution Key Show All Work!!! Page 1 1. (16 points) Circle T if the corresponding statement is True or F if it is False. T F The sequence {1, 1, 1, 1, 1, 1...} is an example of an Alternating sequence.
More information4 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 informationExam 2 Review Chapters 45
Math 365 Lecture Notes S. Nite 8/18/2012 Page 1 of 9 Integers and Number Theory Exam 2 Review Chapters 45 Divisibility Theorem 41 If d a, n I, then d (a n) Theorem 42 If d a, and d b, then d (a+b).
More informationChapter 3: Section 3.1: Factors & Multiples of Whole Numbers
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
More informationApplied Cryptography and Computer Security CSE 664 Spring 2017
Applied Cryptography and Computer Security Lecture 11: Introduction to Number Theory Department of Computer Science and Engineering University at Buffalo 1 Lecture Outline What we ve covered so far: symmetric
More informationSect Complex Numbers
161 Sect 10.8  Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a
More information1 Paid Copy Don t Share With Anyone
HCF & LCM Solved Sums 1. The least common multiple of 24, 36, and 40 is A ) 340 b ) 360 c) 230 d) 400 2 24 36 40 => 2 X 2 X 2 X 3 X 1 X 3 X 5 2 12 18 20 => 360 2 6 9 10 3 3 9 5 1 3 5 Ans : 360 2. The LCM
More informationExercises 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 informationCISC102 Winter 2016 Lecture 11 Greatest Common Divisor
CISC102 Winter 2016 Lecture 11 Greatest Common Divisor Consider any two integers, a,b, at least one nonzero. If we list the positive divisors in numeric order from smallest to largest, we would get two
More informationMasters Tuition Center
1 REAL NUMBERS Exercise 1.1 Q.1. Use Euclid s division algorithm to find the HCF of: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 Solution. (i) In 135 and 225, 225 is larger integer. Using Euclid
More informationChapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices. Integers & Algorithms (2.5)
CSE 54 Discrete Mathematics & Chapter 2 (Part 3): The Fundamentals: Algorithms, the Integers & Matrices Integers & Algorithms (Section 2.5) by Kenneth H. Rosen, Discrete Mathematics & its Applications,
More informationIntegers 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 informationMathematics of Cryptography
Modulo arithmetic Fermat's Little Theorem If p is prime and 0 < a < p, then a p 1 = 1 mod p Ex: 3 (5 1) = 81 = 1 mod 5 36 (29 1) = 37711171281396032013366321198900157303750656 = 1 mod 29 (see http://gauss.ececs.uc.edu/courses/c472/java/fermat/fermat.html)
More informationMath League SCASD. Meet #2. Number Theory. Selfstudy Packet
Math League SCASD Meet #2 Number Theory Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements
More informationBasic 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 informationBasic 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 informationCHAPTER 3. Number Theory
CHAPTER 3 Number Theory 1. Factors or not According to Carl Friedrich Gauss (17771855) mathematics is the queen of sciences and number theory is the queen of mathematics, where queen stands for elevated
More informationcse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska
cse547, math547 DISCRETE MATHEMATICS Professor Anita Wasilewska LECTURE 12 CHAPTER 4 NUMBER THEORY PART1: Divisibility PART 2: Primes PART 1: DIVISIBILITY Basic Definitions Definition Given m,n Z, we say
More informationSummer Math Packet for Students Entering 6th Grade. Please have your student complete this packet and return it to school on Tuesday, September 4.
Summer Math Packet for Students Entering 6th Grade Please have your student complete this packet and return it to school on Tuesday, September. Work on your packet gradually. Complete one to two pages
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether HampdenSydney College Fri, Feb 8, 2013 Robb T. Koether (HampdenSydney College) Direct Proof Divisibility Fri, Feb 8, 2013 1 / 20 1 Divisibility
More informationThis 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 informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether HampdenSydney College Fri, Feb 7, 2014 Robb T. Koether (HampdenSydney College) Direct Proof Divisibility Fri, Feb 7, 2014 1 / 23 1 Divisibility
More informationNUMBER THEORY AND CODES. Álvaro Pelayo WUSTL
NUMBER THEORY AND CODES Álvaro Pelayo WUSTL Talk Goal To develop codes of the sort can tell the world how to put messages in code (public key cryptography) only you can decode them Structure of Talk Part
More informationIntroduction to Information Security
Introduction to Information Security Lecture 5: Number Theory 007. 6. Prof. Byoungcheon Lee sultan (at) joongbu. ac. kr Information and Communications University Contents 1. Number Theory Divisibility
More informationAugust 15, M1 1.4 Common Factors_Multiples Compacted.notebook. Warm Up MI 36. Jun 20 10:53 AM
Warm Up MI 36 8 14 18 Jun 20 10:53 AM 1 Assignment Jun 20 12:36 PM 2 Practice 7 13 A = bh 7 x 13 91 7 7 A = ½bh ½(7 x 7) ½(49) 24.5 Jun 20 12:36 PM 3 Practice 6 4 8 A=½bh 4 6x8 24 A=bh 4x8 32 4 5 8 8 A=bh
More informationArithmetic. Integers: Any positive or negative whole number including zero
Arithmetic Integers: Any positive or negative whole number including zero Rules of integer calculations: Adding Same signs add and keep sign Different signs subtract absolute values and keep the sign of
More informationCOT 3100 Applications of Discrete Structures Dr. Michael P. Frank
University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank Slides for a Course Based on the Text Discrete Mathematics
More informationand LCM (a, b, c) LCM ( a, b) LCM ( b, c) LCM ( a, c)
CHAPTER 1 Points to Remember : REAL NUMBERS 1. Euclid s division lemma : Given positive integers a and b, there exists whole numbers q and r satisfying a = bq + r, 0 r < b.. Euclid s division algorithm
More informationCOMP239: 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 informationIntermediate Math Circles February 14, 2018 Contest Prep: Number Theory
Intermediate Math Circles February 14, 2018 Contest Prep: Number Theory Part 1: Prime Factorization A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. An integer
More informationSection 4. Quantitative Aptitude
Section 4 Quantitative Aptitude You will get 35 questions from Quantitative Aptitude in the SBI Clerical 2016 Prelims examination and 50 questions in the Mains examination. One new feature of the 2016
More informationnot to be republished NCERT REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results
REAL NUMBERS CHAPTER 1 (A) Main Concepts and Results Euclid s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 r < b. Euclid s Division
More informationQ 1 Find the square root of 729. 6. Squares and Square Roots Q 2 Fill in the blank using the given pattern. 7 2 = 49 67 2 = 4489 667 2 = 444889 6667 2 = Q 3 Without adding find the sum of 1 + 3 + 5 + 7
More information{ independent variable some property or restriction about independent variable } where the vertical line is read such that.
Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with
More informationPark Forest Math Team. Meet #2. Number Theory. Selfstudy Packet
Park Forest Math Team Meet #2 Number Theory Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements
More information3 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 informationGRE Quantitative Reasoning Practice Questions
GRE Quantitative Reasoning Practice Questions y O x 7. The figure above shows the graph of the function f in the xyplane. What is the value of f (f( ))? A B C 0 D E Explanation Note that to find f (f(
More informationNumber 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 informationMTH310 EXAM 2 REVIEW
MTH310 EXAM 2 REVIEW SA LI 4.1 Polynomial Arithmetic and the Division Algorithm A. Polynomial Arithmetic *Polynomial Rings If R is a ring, then there exists a ring T containing an element x that is not
More informationMATHEMATICS IN EVERYDAY LIFE 8
MATHEMATICS IN EVERYDAY LIFE Chapter : Square and Square Roots ANSWER KEYS EXERCISE.. We know that the natural numbers ending with the digits,, or are not perfect squares. (i) ends with digit. ends with
More informationNumber Tree LCM HCF Divisibility Rules Power cycle Remainder Theorem Remainder of powers a n b n Last and Second last digit Power of Exponents Euler s
Vedic Numbers Number Tree LCM HCF Divisibility Rules Power cycle Remainder Theorem Remainder of powers a n b n Last and Second last digit Power of Exponents Euler s Theorem Fermet s Theory Wilson Theorem
More informationAlgebra Summer Review Packet
Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills
More informationPrimes and Modular Arithmetic! CSCI 2824, Fall 2014!!
Primes and Modular Arithmetic! CSCI 2824, Fall 2014!!! Scheme version of the algorithm! for finding the GCD (define (gcd a b)! (if!(= b 0)!!!!a!!!!(gcd b (remainder a b))))!! gcd (812, 17) = gcd(17, 13)
More informationEDULABZ INTERNATIONAL NUMBER SYSTEM
NUMBER SYSTEM 1. Find the product of the place value of 8 and the face value of 7 in the number 7801. Ans. Place value of 8 in 7801 = 800, Face value of 7 in 7801 = 7 Required product = 800 7 = 00. How
More informationFACTORS AND MULTIPLES
FACTORS AND MULTIPLES.(A) Find the prime factors of : (i) (ii) (iii) Ans. (i) (ii) (B.) If P n means prime  factors of n, find : (i) P (ii) P (iii) P (iv) P Ans. (i) F =,,, P (Prime factor of ) = and.
More informationUnit 1. Number Theory
Unit 1 Number Theory 11 Divisibility Rules Divisible by: Rule 2 The number is even (it ends in 0, 2, 4, 6 or 8) 3 The sum of its digits is divisible by 3 (eg 741: 7 + 4 + 1 = 12) 4 The last two digits
More informationMathematics 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 informationExecutive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics:
Executive Assessment Math Review Although the following provides a review of some of the mathematical concepts of arithmetic and algebra, it is not intended to be a textbook. You should use this chapter
More informationIntermediate Math Circles Number Theory II Problems and Solutions
WWW.CEMC.UWATERLOO.CA The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Intermediate Math Circles Number Theory II Problems and Solutions 1. The difference between the gcd and lcm of the numbers 10,
More informationMaths Book Part 1. By Abhishek Jain
Maths Book Part 1 By Abhishek Jain Topics: 1. Number System 2. HCF and LCM 3. Ratio & proportion 4. Average 5. Percentage 6. Profit & loss 7. Time, Speed & Distance 8. Time & Work Number System Understanding
More informationChapter 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 informationSolutions to Assignment 1
Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive
More informationAlgebra Introduction to Polynomials
Introduction to Polynomials What is a Polynomial? A polynomial is an expression that can be written as a term or a sum of terms, each of which is the product of a scalar (the coefficient) and a series
More informationPrime Factorizaon. Algebra with Career Applicaons. Find GCF using Prime Factorizaon. Greatest Common Factor. Unit7Vol1.notebook.
Prime Factorizaon Algebra with Career Applicaons Blitzer 5.1 Prime Factorizaon, GCF, LCM A prime number is a number that is divisible by 1 and itself. Examples of primes are 2, 3, 5, 7, 11, 13, 17, 19,
More informationMidterm 1. Your Exam Room: Name of Person Sitting on Your Left: Name of Person Sitting on Your Right: Name of Person Sitting in Front of You:
CS70 Discrete Mathematics and Probability Theory, Fall 2018 Midterm 1 8:0010:00pm, 24 September Your First Name: SIGN Your Name: Your Last Name: Your Exam Room: Name of Person Sitting on Your Left: Name
More informationKnow the Wellordering 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 informationChapter 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 informationNotes 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 informationBefore we talk about prime numbers, we will spend some time with divisibility because there is
Math 1 5.2 Prime Numbers Before we talk about prime numbers, we will spend some time with divisibility. Definition: For whole numbers A and D, with D 0, if there is a whole number Q such that A = D Q,
More information