Course notes on Number Theory


 Oliver Hill
 2 years ago
 Views:
Transcription
1 Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that computers can only work with whole numbers precisely everything else is just an approximation. So, if all that we are allowed to work with are whole numbers, what can we do? Can we add two whole numbers? Yes, because we always get back another whole number. Can we multiply? Yes. Can we divide? No. More often than not, when we divide two whole numbers what we get is a fraction. Therefore, in Number Theory, we ll have to treat division differently. The thing to remember throughout this discussion is that we are trying to figure out how to replace the division operation with the division algorithm. 1 Integer Division What dividing a by b really tells you is how many times you can fit b into a. For example, 2 goes into times. (i.e. 3 2 = 11 ) In general, when 2 we want to know how many times b goes into a, we can always write a in terms of b like so: a = qb + r, where 0 r < b This is known as the division algorithm and it tells us that b goes into a q times with r left over. So, to be formal: Theorem 1 (The Division Algorithm) If a, b Z, with b > 0 then there exist unique q, r Z with a = qb + r, 0 r < b. The number q is called the quotient and r is called the remainder. The division algorithm replaces the division operation in Number Theory as a way to know exactly how b goes into a. Without proving that you can always do this with two such numbers a and b, there are a couple of ways to obtain the equation a = qb + r which can clearly be done regardless of the numbers chosen. One way would be to subtract b from a (or add b to a if a < 0) as many times as possible without going less than zero (or greater than zero if a < 0). Count how many times you can perform the subtraction and that s q. The leftover bit is r. For example, given a = 55 and b = 12, we can do something like this: = = 31 1
2 31 12 = = 7 We performed the subtraction 4 times, and were left with 7 at the end, so 55 = You could also do long division like in grammar school. The number you get above the division bar is q and the remainder is r. (What happens in the first case if b 0?) Once again, the division algorithm replaces the division operation in Number Theory as a way to know exactly how b goes into a. We separate the case where the remainder r is equal to zero. If r = 0 then a = qb, indicating that b goes into a evenly, and we introduce a special notation: Definition 1 We say that b divides a if a = qb for some integer q. We write b a, and say that b is a factor (also called divisor) of a, and a is a multiple of b. 1.1 Some comments on Integer Division Note that in the last definition we say that the statement (b a) is equivalent to (a = qb) but not to ( a = q). This is because in Number Theory, there is b no division operator, and a has no meaning...technically. This isn t to say b that you are wrong to assume that a = q when you are told that b a, it b just means that you are now working outside Number Theory. As an example of how the division algorithm can replace the division operation, think of how a computer evaluates the expression a / b when a and b are both integers. The / operator always returns an integer, even when b does not divide a. The integer that it returns is the same q from a = qb + r. So what you thought was the division operator for integer type variables is really just an application of the division algorithm. It should also be mentioned that the division algorithm is not really an algorithm the equation which is guaranteed to exist by Theorem 1 is the result of an algorithm. 2 Primes and Composites Recall that if b a, then b is a factor of a. For example, the set of all of the factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}. 2
3 Definition 2 For any integer n > 1, n has at least two factors, namely itself and 1. An integer p > 1 is called prime if it has itself and 1 as its only factors. Otherwise, it is called composite The set of primes, {2, 3, 5, 7, 11, 13, 17, 23,...} is countably infinite. It is not too difficult to imagine that, if you write n as the product of two of its factors x and y, then rewrite x and y each as the product of two of their factors, and so on, you may get to the point where one of the factors in the product is prime and can t be rewritten differently (except as itself times 1). In fact, if you do this, you will always eventually get to the point where all of the factors are prime, and you cannot decompose the number n any further. This is called the prime factorization of n For example: 24 = 6 4 = (3 2) (2 2) = What if instead of starting with 24 = 6 4, we wrote 24 = 8 3? Would we end up with the same prime factorization? 24 = 8 3 = (4 2) 3 = ((2 2) 2) 3 = In fact, we will always end up with same prime factorization of an integer n > 1, no matter what n is, and no matter in what way we decide to decompose n. This is stated precisely by the Fundamental Theorem of Arithmetic. Theorem 2 (The Fundamental Theorem of Arithmetic) Every integer n > 1 can be written as a product of primes uniquely, up to the order of the primes. This means that any number can be written uniquely as n = p n 1 1 pn 2 2 pn pn k k..., where p i represents the i th prime number (i.e. p 1 = 2, p 2 = 3, p 3 = 5, etc.) and all the n i are integers greater than or equal to zero. If, for example, you have an expression A = B, then A = p A 1 1 pa 2 2 pa p A k k..., and B = p B 1 1 pb 2 2 pb p B k k..., and by the Fundamental Theorem of Arithmetic, you can conclude that i, A i = B i 3
4 It is convenient to write the generalized prime factorization in its compact form: n = p n i i = p n 1 1 pn 2 2 pn pn k k..., i=1 Even though the product goes until infinity, there is some point after which all n i = 0. 3 Greatest Common Divisor and Least Common Multiple These terms should sound familiar. Definition 3 (Greatest Common Divisor) Let a and b be integers which are not both zero. The largest integer d that divides both a and b is denoted by gcd(a, b). gcd(a, b) is a function from Z Z Z and d = gcd(a, b) It should be noted that gcd(a, 0) = a, and you should justify to yourself why that satisfies the definition. You may be familiar with the greatest common divisor from arithmetic. When you need to reduce a fraction m n, you divide top and bottom by gcd(m, n). For example, to reduce the fraction: = = = = 5 14 you divide the numerator and denominator by = 120 = gcd(600, 1680). It shouldn t be too difficult to see that gcd(m, n) = i=1 p min(m i,n i ) i Definition 4 (Least Common Divisor) Let a and b be positive integers. The smallest positive integer d that can be divided by both a and b is denoted by lcm(a, b). lcm(a, b) is a function from N N N and d = lcm(a, b) The least common multiple arises when you take the sum of two fractions. The denominator of the sum of two fractions 1 x and 1 will always be y lcm(x, y) if you reduce the answer to lowest terms. For example: 4
5 = = = ( ) ( ) = (23 3 5) ( ) ( ) ( ) = From this you should see that ( ) ( ) = lcm = i=1 p max(m i,n i ) i You may also notice from the fifth equation of the example that a b = gcd(a, b) lcm(a, b) and that to pull out a common factor from (A + B) you can pull out all common factors by (A + B) = gcd(a, B) ( A gcd(a, B) + B ) gcd(a, B) 3.1 The Euclidean Algorithm The Euclidean Algorithm is a slick little number that can be used to find the gcd of very large numbers relatively quickly, and it depends on the following theorem: Theorem 3 Let a = qb + r, then gcd(a, b) = gcd(b, r) Proof: Denote the set S (a,b) = {d (d a) (d b)} be the set of all common divisors of a and b. Likewise, the set S (b,r) is the set of all common divisors of b and r. If we can show that S (a,b) = S (b,r), then we ll have shown that gcd(a, b) = gcd(b, r) since the largest element of S (a,b) must also be the largest element of S (a,b). So, if d S (a,b) then d a and d b. If d b then d qb. If d qb and d a then d (a qb) 5
6 Since a = qb + r, a qb = r, therefore, d r So, d b and d r therefore, d S (b,r) If d S (b,r), then d b and d r. If d b then d qb. If d qb and d r then d (qb + r) Since a = qb + r, d a So, d a and d b therefore, d S (a,b) Therefore, S (a,b) = S (b,r) and it follows that gcd(a, b) = gcd(b, r) So, to perform the Euclidean Algorithm to find the gcd(a, b), (where a > b) write a = q 1 b + r 1 and you ll have gcd(a, b) = gcd(b, r). Now write b = q 2 r 1 + r 2 to get gcd(a, b) = gcd(b, r 1 ) = gcd(r 1, r 2 ), and so on, until you get down to the gcd(a, b) = gcd(b, r 1 ) = gcd(r 1, r 2 ) =... = gcd(r n 1, r n ) = gcd(r n, 0) = r n. This is best understood by example, of which there are plenty in the book. After you ve done a couple, you ll be an expert. An even simpler way is as follows: Write (a, b). Underneath that, write (a b, b) if a > b or (a, b a) if b > a. Repeat until you get to (x, x). Then gcd(a, b) = x. Here s an example, solving the same problem as in Example 1 on page 129 of the textbook. (414, 662) (414, 248) (166, 248) (166, 82) (84, 82) (2, 82) (2, 80) (2, 78). (2, 2) Therefore, gcd(414, 662) = 2. In fact, feel free at any point in the algorithm to stop if it is obvious what the gcd is. For example, at (2, 82) above, it is clear that gcd(2, 82) = 2. 6
7 4 Modular Arithmetic All of this development of Number Theory really leads up to this. The way that we ve treated division is so cool, that we can actually develop an entire system of arithmetic based on the remainder r from a = qb + r. We do this by defining an operator and a relation. Definition 5 (Mod Operator) Let the operator mod, or % which takes an an integer a and a natural number m be defined as amodm = r (or a%m = r) where r is the remainder after division of a by m. With this definition, we could rewrite the division algorithm as a = qm + amodm. Please note that, just as in the division algorithm, m must be a positive integer, but a can be anything. Now the relation: Definition 6 (Congruency Modulo m) Two numbers a and b are congruent modulo m iff amodm = bmodm. We write a b (mod m). Please study these definitions and understand their difference as well as their similarity. A couple of thoughts on the congruency relation. First, it is important to note that what we have just defined is the part. The (mod m) part really just refers to the symbol, letting the reader know which equivalence relation is being talked about. It may be helpful to realize that there are some who write a is congruent to b modulo m like a m b. Perhaps this makes it easier to understand that a b (mod 3) is not the same as a b (mod 5). We ve defined a whole family of relations which depend on which m you pick. For example, 15 is congruent to 8, 22, 6 and 13 modulo 7 (they all have remainder 1 when divided by 7), but it s not congruent to any of them modulo 5. The following 5 statements about integers are all equivalent: 1. a b (mod m) 2. a b m = k 3. m (a b) 4. a mod m = b mod m or a%m = b%m 5. a = mk + b 7
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 informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationCLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010
CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 1. Greatest common divisor Suppose a, b are two integers. If another integer d satisfies d a, d b, we call d a common divisor of a, b. Notice that as
More informationThe Euclidean Algorithm
The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have
More informationToday s Topics. Primes & Greatest Common Divisors
Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime
More informationFractions and Decimals
Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
More informationCHAPTER 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
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More informationMathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR
Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGrawHill The McGrawHill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,
More informationPreAlgebra Class 3  Fractions I
PreAlgebra Class 3  Fractions I Contents 1 What is a fraction? 1 1.1 Fractions as division............................... 2 2 Representations of fractions 3 2.1 Improper fractions................................
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationMATH 13150: Freshman Seminar Unit 10
MATH 13150: Freshman Seminar Unit 10 1. Relatively prime numbers and Euler s function In this chapter, we are going to discuss when two numbers are relatively prime, and learn how to count the numbers
More informationMath 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
More informationGreatest Common Divisor
Greatest Common Divisor Blake Thornton Much of this is due directly to Josh Zucker, Paul Zeitz and Harold Reiter. Divisibility 1. How can 48 be built from primes? In what sense is there only one way? How
More informationa = bq + r where 0 r < b.
Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental
More informationFactoring Whole Numbers
2.2 Factoring Whole Numbers 2.2 OBJECTIVES 1. Find the factors of a whole number 2. Find the prime factorization for any number 3. Find the greatest common factor (GCF) of two numbers 4. Find the GCF for
More informationAdding and Subtracting Fractions. 1. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.
Tallahassee Community College Adding and Subtracting Fractions Important Ideas:. The denominator of a fraction names the fraction. It tells you how many equal parts something is divided into.. The numerator
More informationApplications of Fermat s Little Theorem and Congruences
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
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationSometimes it is easier to leave a number written as an exponent. For example, it is much easier to write
4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More informationStupid 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
More informationDiscrete 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
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L34) is a summary BLM for the material
More informationElementary 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
More informationCOMPASS Numerical Skills/PreAlgebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13
COMPASS Numerical Skills/PreAlgebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
More informationNow that we have a handle on the integers, we will turn our attention to other types of numbers.
1.2 Rational Numbers Now that we have a handle on the integers, we will turn our attention to other types of numbers. We start with the following definitions. Definition: Rational Number any number that
More informationIntegers and division
CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.
More informationNumber 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
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationV55.0106 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
More informationHomework 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
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More informationIntroduction to Diophantine Equations
Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations
More informationA.4 Polynomial Division; Synthetic Division
SECTION A.4 Polynomial Division; Synthetic Division 977 A.4 Polynomial Division; Synthetic Division OBJECTIVES 1 Divide Polynomials Using Long Division 2 Divide Polynomials Using Synthetic Division 1 Divide
More informationThe Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006
The Teacher s Circle Number Theory, Part 1 Joshua Zucker, August 14, 2006 joshua.zucker@stanfordalumni.org [A few words about MathCounts and its web site http://mathcounts.org at some point.] Number theory
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationMATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS
MATH 537 (Number Theory) FALL 2016 TENTATIVE SYLLABUS Class Meetings: MW 2:003:15 pm in Physics 144, September 7 to December 14 [Thanksgiving break November 23 27; final exam December 21] Instructor:
More informationPreviously, you learned the names of the parts of a multiplication problem. 1. a. 6 2 = 12 6 and 2 are the. b. 12 is the
Tallahassee Community College 13 PRIME NUMBERS AND FACTORING (Use your math book with this lab) I. Divisors and Factors of a Number Previously, you learned the names of the parts of a multiplication problem.
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationDefinition 1 Let a and b be positive integers. A linear combination of a and b is any number n = ax + by, (1) where x and y are whole numbers.
Greatest Common Divisors and Linear Combinations Let a and b be positive integers The greatest common divisor of a and b ( gcd(a, b) ) has a close and very useful connection to things called linear combinations
More informationSession 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
More informationCONTENTS. Please note:
CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division
More informationGrade 6 Math Circles March 10/11, 2015 Prime Time Solutions
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going
More informationHandout 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
More information6 Proportion: Fractions, Direct and Inverse Variation, and Percent
6 Proportion: Fractions, Direct and Inverse Variation, and Percent 6.1 Fractions Every rational number can be written as a fraction, that is a quotient of two integers, where the divisor of course cannot
More informationCISC  Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association
CISC  Curriculum & Instruction Steering Committee California County Superintendents Educational Services Association Primary Content Module IV The Winning EQUATION NUMBER SENSE: Factors of Whole Numbers
More informationIntermediate Math Circles March 7, 2012 Linear Diophantine Equations II
Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II Last week: How to find one solution to a linear Diophantine equation This week: How to find all solutions to a linear Diophantine
More information3. Applications of Number Theory
3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a
More informationGreatest Common Factor and Least Common Multiple
Greatest Common Factor and Least Common Multiple Intro In order to understand the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM), we need to define two key terms: Multiple: Multiples
More informationGCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!
GCDs and Relatively Prime Numbers! CSCI 2824, Fall 2014!!! Challenge Problem 2 (Mastermind) due Fri. 9/26 Find a fourth guess whose scoring will allow you to determine the secret code (repetitions are
More informationKevin 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
More informationMultiplying and Dividing Fractions
Multiplying and Dividing Fractions 1 Overview Fractions and Mixed Numbers Factors and Prime Factorization Simplest Form of a Fraction Multiplying Fractions and Mixed Numbers Dividing Fractions and Mixed
More informationIntroduction to Fractions
Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying
More informationClock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system
CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving NonConditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationU.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
More information4.4 Clock Arithmetic and Modular Systems
4.4 Clock Arithmetic and Modular Systems A mathematical system has 3 major properies. 1. It is a set of elements 2. It has one or more operations to combine these elements (ie. Multiplication, addition)
More informationRevised 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.
More information1. The Fly In The Ointment
Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 20122013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More informationa 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
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationDecimal Notations for Fractions Number and Operations Fractions /4.NF
Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.
More informationPrime Factorization 0.1. Overcoming Math Anxiety
0.1 Prime Factorization 0.1 OBJECTIVES 1. Find the factors of a natural number 2. Determine whether a number is prime, composite, or neither 3. Find the prime factorization for a number 4. Find the GCF
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationMATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003
MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld
More information3.4 Complex Zeros and the Fundamental Theorem of Algebra
86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and
More informationIntroduction to Fractions
Introduction to Fractions Fractions represent parts of a whole. The top part of a fraction is called the numerator, while the bottom part of a fraction is called the denominator. The denominator states
More informationNumber 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
More information17 Greatest Common Factors and Least Common Multiples
17 Greatest Common Factors and Least Common Multiples Consider the following concrete problem: An architect is designing an elegant display room for art museum. One wall is to be covered with large square
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationGREATEST 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
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem Evan Chen evanchen@mit.edu February 3, 2015 The Chinese Remainder Theorem is a theorem only in that it is useful and requires proof. When you ask a capable 15yearold why
More informationABSTRACT 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
More informationLesson 4. Factors and Multiples. Objectives
Student Name: Date: Contact Person Name: Phone Number: Lesson 4 Factors and Multiples Objectives Understand what factors and multiples are Write a number as a product of its prime factors Find the greatest
More informationQuotient 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.
More informationAccuplacer Arithmetic Study Guide
Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationDecimals and other fractions
Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very
More informationAn Introduction to Number Theory Prime Numbers and Their Applications.
East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 82006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRETEST
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationUseful Number Systems
Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationA Prime Investigation with 7, 11, and 13
. Objective To investigate the divisibility of 7, 11, and 13, and discover the divisibility characteristics of certain sixdigit numbers A c t i v i t y 3 Materials TI73 calculator A Prime Investigation
More informationGrade 7/8 Math Circles Fall 2012 Factors and Primes
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Factors and Primes Factors Definition: A factor of a number is a whole
More informationnorth seattle community college
INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The
More informationModule 2: Working with Fractions and Mixed Numbers. 2.1 Review of Fractions. 1. Understand Fractions on a Number Line
Module : Working with Fractions and Mixed Numbers.1 Review of Fractions 1. Understand Fractions on a Number Line Fractions are used to represent quantities between the whole numbers on a number line. A
More informationIntegers, Division, and Divisibility
Number Theory Notes (v. 4.23 October 31, 2002) 1 Number Theory is the branch of mathematics that deals with integers and their properties, especially properties relating to arithmetic operations like addition,
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More information