Euler Enrichment Stage
|
|
|
- Sherilyn Patrick
- 9 years ago
- Views:
Transcription
1 Euler Enrichment Stage Table of Contents Chapter 1 Problems I Like Michael Evans 1 Chapter 2 Primes and Composites 3 Chapter 3 Least Common Multiple 8 Chapter 4 Highest Common Factor and the Euclidean Algorithm 11 Chapter 5 Arithmetic Sequences 16 Chapter 6 Figurate Numbers 21 Chapter 7 Congruences 26 Chapter 8 Problems I Like Sharing - Alice Thomas 31 Chapter 9 Find That Angle 34 Chapter 10 Counting Techniques 51 Chapter 11 The Pigeon-Hole Principle 58 Chapter 12 Problems I Enjoy Sharing - Bruce Henry 62 Solutions 64
2 Chapter 1 Problems I Like Michael Evans 1 In magic squares, the sum of the numbers in each row, each column and each diagonal is constant Find A, B, C, D, E in the following magic square 15 A B C 25 D E 2 (a) What two whole numbers, neither ending in zero, when multiplied together equal exactly ? (b) Repeat for Find three consecutive numbers such that the sum of the first and third is 18 4 A palindromic number is a number which remains the same when the digits are reversed For example, is a palindromic number What is the next largest palindromic number? 5 What two-digit number is twice the product of its digits? 6 Find the values of the letters, each of which stands for a particular but different digit + + F O R T Y T E N T E N S I X T Y 7 By replacing the asterisks with a selection of the four operational symbols (+,,, ), complete this equation to make a true statement: =13 8 A prime number is an integer greater than 1 whose only divisors are itself and is a prime What is the next year that is a prime? 1
3 9 What is the largest three-digit prime each of whose digits is a prime? 10 A perfect number equals the sum of its factors, excluding the number itself Since 6 = , 6 is a perfect number Find another 11 Divide the face of the clock into 3 parts with 2 straight lines so that the sums of the numbers in the 3 parts are equal A total of 642 digits was used in numbering the pages of a book How many pages did the book contain? 13 In the cells shown, place a ten-digit number such that the digit in the first cell indicates the total number of 0s in the entire number, the digit in cell 1 indicates the number of 1s in the number and so on, to the last cell Now try Problem 1 in the Euler Student Problems Book 2
4 Chapter 2 Primes and Composites A code is devised by Secret Sam Sam has decided that he will denote A by the smallest number with 1 factor, B the smallest number with 2 factors and so on He sets out his code as shown in the table (a) (b) A 1 B 2 C 4 D 6 E 16 Z Complete the table for the code Secret Sam decides that he would like to punctuate his code properly and use some mathematical symbols For example he denotes = with the smallest number with 39 factors What is this number? Factors and Primes The factors of 6 are: 1, 2, 3 and 6 The factors of 28 are: 1, 2, 4, 7, 14 and 28 The factors of 7 are: 1 and 7 A natural number, a, isafactor of a natural number, b, ifthereexistsa natural number, k, such that: b = ak If a number greater than 1 has only factors 1 and itself it is said to be a prime Among the first 100 numbers, the following 25 are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 To consider whether a number, n, isprimeonemethodistoconsiderall the numbers less than n and test if they are factors of n For example, if n = 97, 9 < n<10 Therefore, we could test the numbers: 2, 3, 4, 5, 6, 7, 8 and 9 However, we do not need to test numbers which are not prime, so we only need to test 2, 3, 5 and 7 3
5 It is found that none of these numbers is a factor of 97 We have found 97 to be prime There are infinitely many primes Composites Anumber,m, is called a composite number if it can be written as a product: m = a b where a and b are numbers greater than 1 and less than m For example, 72 = 8 9= = = = = = This method of expressing a composite number in terms of its prime numbers is called prime decomposition For each number there is a unique prime decomposition One method of determining the prime decomposition is the following The process is repeated division Consider the number Therefore = The number of factors of a number can be determined through the prime decomposition of that number They can be systematically listed as shown: 4
6 This covers all combinations Thus there are = 64 factors of Alternatively, we note that the prime decomposition of is and that 2 7 has 8 factors, 3 3 has4factorsand7 1 has 2 factors Hence has factors This is investigated further in the Exercises of this chapter Exercises 1 Give the prime decomposition of Give the prime decomposition of (a) State the factors of 16 = 2 4 (note: include 1 and 16) (b) How many factors are there of 16? (c) State the factors of 256 = 2 8 (d) How many factors are there of 256? (e) State the number of factors of 2 n? 4 (a) State the factors of 125 = 5 3 (b) How many factors are there of 5 n? 5 (a) Give the prime decomposition of (b) Howmanyfactorsdoesthisnumberhave? 6 A number has a prime decomposition of How many factors does this number have? 7 Two and three differ by one and are both prime numbers What is the next such pair? 8 Find a prime number which is one less than a perfect square Is there another such number? 5
Homework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
Grade 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
Session 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
= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
Chapter 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
. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
Just the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
CONTENTS. 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
Cubes and Cube Roots
CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy
Chapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
Factoring 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
ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite
ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,
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.
Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7 & 8 Math Circles October 19, 2011 Prime Numbers Factors Definition: A factor of a number is a whole
Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
Category 3 Number Theory Meet #1, October, 2000
Category 3 Meet #1, October, 2000 1. For how many positive integral values of n will 168 n be a whole number? 2. What is the greatest integer that will always divide the product of four consecutive integers?
Working with whole numbers
1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and
Grade 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
SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
CHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
MACM 101 Discrete Mathematics I
MACM 101 Discrete Mathematics I Exercises on Combinatorics, Probability, Languages and Integers. Due: Tuesday, November 2th (at the beginning of the class) Reminder: the work you submit must be your own.
Previously, 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.
Prime 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
CISC - 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
5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
Clock 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
FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M.
Mathematics Revision Guides Factors, Prime Numbers, H.C.F. and L.C.M. Page 1 of 16 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
Recall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points.
2 MODULE 4. DECIMALS 4a Decimal Arithmetic Adding Decimals Recall the process used for adding decimal numbers. Adding Decimals. To add decimal numbers, proceed as follows: 1. Place the numbers to be added
k, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
Some practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a long-winded way of asking for the value of 6 2000 mod
SOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
Applications 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
Today 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
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
MATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
SECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
Notes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
Primes. Name Period Number Theory
Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in
1. When the least common multiple of 8 and 20 is multiplied by the greatest common factor of 8 and 20, what is the result?
Black Equivalent Fractions and LCM 1. When the least common multiple of 8 and 20 is multiplied by the greatest common factor of 8 and 20, what is the result? 2. The sum of three consecutive integers is
Pigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
MATHEMATICS. Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples. Equipment. MathSphere
MATHEMATICS Y5 Multiplication and Division 5330 Square numbers, prime numbers, factors and multiples Paper, pencil, ruler. Equipment MathSphere 5330 Square numbers, prime numbers, factors and multiples
Continued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
Factoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
15 Prime and Composite Numbers
15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such
4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
3.1. 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
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
The 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
Grade 6 Math Circles. Binary and Beyond
Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number
Integer Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 [email protected] March 16, 2011 Abstract We give
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
CS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
DigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
SUM OF TWO SQUARES JAHNAVI BHASKAR
SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted
We can express this in decimal notation (in contrast to the underline notation we have been using) as follows: 9081 + 900b + 90c = 9001 + 100c + 10b
In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should
Math 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
The application of prime numbers to RSA encryption
The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 50 51. 2
1. Use Euler s trick to find the sum 1 + 2 + 3 + 4 + + 49 + 50. s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 Thus, 2s = 50 51. Therefore, s = 50 51. 2 2. Consider the sequence
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks
MATHCOUNTS TOOLBOX Facts, Formulas and Tricks MATHCOUNTS Coaching Kit 40 I. PRIME NUMBERS from 1 through 100 (1 is not prime!) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 II.
POLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
Teaching & Learning Plans. Arithmetic Sequences. Leaving Certificate Syllabus
Teaching & Learning Plans Arithmetic Sequences Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.
Lecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
Sample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY
PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences
Not for resale. 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples
CHAPTER 4 Number Theory 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples 4.4 Codes and Credit Card Numbers: Connections to Number Theory
Playing with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
MATH 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
Continued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
4.5 Finite Mathematical Systems
4.5 Finite Mathematical Systems We will be looking at operations on finite sets of numbers. We will denote the operation by a generic symbol, like *. What is an operation? An operation is a rule for combining
Polynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
The last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
Patterns in Pascal s Triangle
Pascal s Triangle Pascal s Triangle is an infinite triangular array of numbers beginning with a at the top. Pascal s Triangle can be constructed starting with just the on the top by following one easy
Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
Solution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.
[Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write
FACTORS AND MULTIPLES Answer Key
I. Find prime factors by factor tree method FACTORS AND MULTIPLES Answer Key a. 768 2 384 2 192 2 96 2 48 2 24 2 12 2 6 2 3 768 = 2*2*2*2*2*2*2*2 *3 b. 1608 3 536 2 268 2 134 2 67 1608 = 3*2*2*2*67 c.
CS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a
IB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
Multiplication. Year 1 multiply with concrete objects, arrays and pictorial representations
Year 1 multiply with concrete objects, arrays and pictorial representations Children will experience equal groups of objects and will count in 2s and 10s and begin to count in 5s. They will work on practical
CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: [email protected] URL: http://home.imf.au.dk/niels/ Contents
A Prime Investigation with 7, 11, and 13
. Objective To investigate the divisibility of 7, 11, and 13, and discover the divisibility characteristics of certain six-digit numbers A c t i v i t y 3 Materials TI-73 calculator A Prime Investigation
Greatest Common Factors and Least Common Multiples with Venn Diagrams
Greatest Common Factors and Least Common Multiples with Venn Diagrams Stephanie Kolitsch and Louis Kolitsch The University of Tennessee at Martin Martin, TN 38238 Abstract: In this article the authors
10/14/08 ON NUMBERS EQUAL TO THE SUM OF TWO SQUARES IN MORE THAN ONE WAY
1 10/14/08 ON NUMBERS EQUAL TO THE SUM OF TWO SQUARES IN MORE THAN ONE WAY Ming-Sun Li, Kathryn Robertson, Thomas J. Osler, Abdul Hassen, Christopher S. Simons and Marcus Wright Department of Mathematics
A Study on the Necessary Conditions for Odd Perfect Numbers
A Study on the Necessary Conditions for Odd Perfect Numbers Ben Stevens U63750064 Abstract A collection of all of the known necessary conditions for an odd perfect number to exist, along with brief descriptions
How To Understand And Solve Algebraic Equations
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20.
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
Systems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
IB Math Research Problem
Vincent Chu Block F IB Math Research Problem The product of all factors of 2000 can be found using several methods. One of the methods I employed in the beginning is a primitive one I wrote a computer
26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue