# Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system

Size: px
Start display at page:

Download "Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system"

Transcription

1 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 system as () a set of elements, along with () one or more operations for combining those elements, and () one or more relations for comparing those elements. We now consider the -hour clock system, which is based on an ordinary clock face, except that is replaced with and only a single hand, say the hour hand, is used. See Figure. The clock face yields the finite set {,,,,,,,,,,, }. As an operation for this clock system, addition is defined as follows: add by moving the hour hand in a clockwise direction. For example, to add and on a clock, first move the hour hand to, as in Figure. Then, to add, move the hour hand more hours in a clockwise direction. The hand stops at, so. This result agrees with traditional addition. However, the sum of two numbers from the -hour clock system is not always what might be expected, as the following example shows. EXAMPLE Find each sum in -hour clock arithmetic. (a) Move the hour hand to, as in Figure. Then advance the hand clockwise through more hours. It stops at, so. (b) Proceed as shown in Figure. Check that. Since there are infinitely many whole numbers, it is not possible to write a complete table of addition facts for that set. Such a table, to show the sum of every possible pair of whole numbers, would have an infinite number of rows and columns, making it impossible to construct. On the other hand, the -hour clock system uses only the whole numbers,,,,,,,,,,, and. A table of all possible sums for this system requires only rows and columns. The -hour clock addition table is shown in Table. Since the -hour system is built upon a finite set, it is called a finite mathematical system. TABLE -Hour Clock Addition

2 . Clock Arithmetic and Modular Systems EXAMPLE Use the -hour clock addition table to find each sum. Plagued by serious maritime mishaps linked to navigational difficulties, several European governments offered prizes for an effective method of determining longitude. The largest prize was, pounds (equivalent to several million dollars in today s currency) offered by the British Parliament in the Longitude Act of. While famed scientists, academics, and politicians pursued an answer in the stars, John Harrison, a clock maker, set about to build a clock that could maintain accuracy at sea. This turned out to be the key, and Harrison s Chronometer eventually earned him the prize. For a fascinating account of this drama and of Harrison s struggle to collect his prize money from the government, see the book The Illustrated Longitude by Dava Sobel and William J. H. Andrewes. (a) Find on the left of the addition table and across the top. The intersection of the row headed and the column headed gives the number. Thus,. (b) Also from the table,. So far, our -hour clock system consists of the set {,,,,,,,,,,, }, together with the operation of clock addition. Next we will check whether this sytem has the closure, commutative, associative, identiy, and inverse properties, as described in Section.. Table shows that the sum of two numbers on a clock face is always a number on the clock face. That is, if a and b are any clock numbers in the set of the system, then a b is also in the set of the system. The system has the closure property. (The set of the system is closed under clock addition.) Notice also that in this system, and both yield. Also and both yield. The order in which elements are added does not seem to matter. In fact, you can see in Table that the part of the table above the colored diagonal line is a mirror image of the part below the diagonal. This shows that, for any clock numbers a and b, a b b a. The system has the commutative property. The next question is: When three elements are combined in a given order, say a b c, does it matter whether the first and second or the second and third are associated initially? In other words, is it true that, for any a, b, and c in the system, a b c a b c? EXAMPLE Is -hour clock addition associative? It would take lots of work to prove that the required relationship always holds. But a few examples should either disprove it (by revealing a counterexample a case where it fails to hold), or should make it at least plausible. Using the clock numbers,, and, we see that. Thus,. Try another example:. So. Other examples also will work. The -hour clock system has the associative property. Our next question is whether the clock face contains some element (number) which, when combined with any element (in either order), produces that same element. Such an element (call it e) would satisfy a e a and e a a for any element a of the system. Notice in Table that,, and so on. The number is the required identity element. The system has the identity property. Generally, if a finite system has an identity element e, it can be located easily in the operation table. Check the body of Table for a column that is identical to the column at the left side of the table. Since the column under meets this requirement, a a holds for all elements a in the system. Thus is possibly the identity.

3 CHAPTER Number Theory Minus hours = FIGURE Now locate at the left of the table. Since the corresponding row is identical to the row at the top of the table, a a also holds for all elements a, which is the other requirement of an identity element. Hence is indeed the identity. Subtraction can be performed on a -hour clock. Subtraction may be interpreted on the clock face by a movement in the counterclockwise direction. For example, to perform the subtraction, begin at and move hours counterclockwise, ending at, as shown in Figure. Therefore, in this system,. In our usual system, subtraction may be checked by addition and this is also the case in clock arithmetic. To check that, simply add. The result on the clock face is, verifying the accuracy of this subtraction. The additive inverse, a, of an element a in clock arithmetic is the element that satisfies the following statement: a a and a a. The next example examines this idea. EXAMPLE Determine the additive inverse of in -hour clock arithmetic. The additive inverse for the clock number is a number x such that x. Going from to on the clock face requires more hours, so. This means that is the additive inverse of. The method used in Example may be used to verify that every element of the system has an additive inverse (also in the system). So the system has the inverse property. A simpler way to verify the inverse property, once you have the table, is to make sure the identity element appears exactly once in each row, and that the pair of elements that produces it also produces it in the opposite order. (This last condition is automatically true if the commutative property holds for the system.) For example, note in Table that row contains one, under the, so, and that row contains, under the, so also. Therefore, and are inverses. The following chart lists the elements and their additive inverses. Notice that one element,, is its own inverse for addition. Clock value a Additive inverse a Using the additive inverse symbol, we can say that in clock arithmetic,,,, and so on. We have now seen that the -hour clock system, with addition, has all five of the potential properties of a single-operation mathematical system, as discussed in Section.. Other operations can also be introduced to the clock arithmetic system.

4 . Clock Arithmetic and Modular Systems Having discussed additive inverses, we can define subtraction formally. Notice that the definition is the same as for ordinary subtraction of whole numbers. Subtraction on a Clock If a and b are elements in clock arithmetic, then the difference, is defined as a b a (b). a b, Chess Clock A double clock is used to time chess, backgammon, and Scrabble games. Push one button, and that clock stops the other begins simultaneously. When a player s allotted time for the game has expired, that player will lose if he or she has not made the required number of moves. Mathematics and chess both involve structured relationships and demand logical thinking. A chess player may depend more on psychological acumen and knowledge of the opponent than on the mathematical component of the game. Emanuel Lasker achieved mastery in both fields. He was best known as a World Chess Champion for years, until. Lasker also was famous in mathematical circles for his work concerning the theory of primary ideals, algebraic analogies of prime numbers. An important result, the Lasker-Noether theorem, bears his name along with that of Emmy Noether. Noether extended Lasker s work. Her father had been Lasker s Ph.D. advisor. EXAMPLE Find each of the following differences. (a) Use the definition of subtraction. The additive inverse of is, from the table of inverses. This result agrees with traditional arithmetic. Check by adding and ; the sum is. (b) Clock numbers can also be multiplied. For example, find the product by adding a total of times:. EXAMPLE Find each product, using clock arithmetic. (a) (b) (c) (d) Some properties of the system of -hour clock numbers with the operation of multiplication will be investigated in Exercises. Modular Systems We now expand the ideas of clock arithmetic to modular systems in general. Recall that -hour clock arithmetic was set up so that answers were always whole numbers less than. For example,. The traditional sum,, reflects the fact that moving the clock hand forward hours from, and then forward another hours, amounts to moving it forward hours total. But since the final position of the clock is at, we see that and are, in a sense, equivalent. More formally we say that and are congruent modulo (or congruent mod ), which is written mod (The sign indicates congruence.)

5 CHAPTER Number Theory By observing clock hand movements, you can also see that, for example, mod, mod, and so on. In each case, the congruence is true because the difference of the two congruent numbers is a multiple of :, This suggests the following definition.,. Congruence Modulo m The integers a and b are congruent modulo m (where m is a natural number greater than called the modulus) if and only if the difference a b is divisible by m. Symbolically, this congruence is written a b (mod m). Since being divisible by m is the same as being a multiple of m, we can say that a b mod m if and only if a b km for some integer k. EXAMPLE Decide whether each statement is true or false. (a) mod The difference is divisible by, so mod is true. (b) mod This statement is false, since, which is not divisible by. (c) mod This statement is true, since is divisible by. (It doesn t matter if we find or.) There is another method of determining if two numbers, a and b, are congruent modulo m. Criterion for Congruence a b mod m if and only if the same remainder is obtained when a and b are divided by m. For example, we know that mod because, which is divisible by. Now, if is divided by, the quotient is and the remainder is. Also, if is divided by, the quotient is and the remainder is. According to the criterion above, mod since both remainders are the same. Addition, subtraction, and multiplication can be performed in any modular system just as with clock numbers. Since final answers should be whole numbers less than the modulus, we can first find an answer using ordinary arithmetic. Then,

6 . Clock Arithmetic and Modular Systems as long as the answer is nonnegative, simply divide it by the modulus and keep the remainder. This produces the smallest nonnegative integer that is congruent (modulo m) to the ordinary answer. EXAMPLE Find each of the following sums, differences, and products. (a) (b) (c) (d) (e) mod First add and to get. Then divide by. The remainder is, so mod and mod. mod. Divide by, obtaining as a remainder: mod. mod. When is divided by, a remainder of is found: mod. mod Since, and leaves a remainder of when divided by, mod. mod mod. Problem Solving Modular systems can often be applied to questions involving cyclical changes. For example, our method of dividing time into weeks causes the days to repeatedly cycle through the same pattern of seven. Suppose today is Sunday and we want to know what day of the week it will be days from now. Since we don t care how many weeks will pass between now and then, we can discard the largest whole number of weeks in days and keep the remainder. (We are finding the smallest nonnegative integer that is congruent to modulo.) Dividing by leaves remainder, so the desired day of the week is days past Sunday, or Wednesday. EXAMPLE If today is Thursday, November, and next year is a leap year, what day of the week will it be one year from today? A modulo system applies here, but we need to know the number of days between today and one year from today. Today s date, November, is unimportant except that it shows we are later in the year than the end of February and therefore the next year (starting today) will contain days. (This would not be so if today were, say, January.) Now dividing by produces with remainder. Two days past Thursday is our answer. That is, one year from today will be a Saturday.

7 CHAPTER Number Theory Problem Solving A modular system (mod m) allows only a fixed set of remainder values,,,,, m. One practical approach to solving modular equations, at least when m is reasonably small, is to simply try all these integers. For each solution found in this way, others can be found by adding multiples of the modulus to it. EXAMPLE Solve x mod. In a mod system, any integer will be congruent to one of the integers,,,,,, or. So, the equation x mod can be solved by trying, in turn, each of these integers as a replacement for x. x : Is it true that mod? No x : Is it true that mod? No x : Is it true that mod? Yes Try x, x, x, and x to see that none of them work. Of the integers from through, only is a solution of the equation x mod. Since is a solution, find other solutions to this mod equation by repeatedly adding :,,, and so on. The set of all positive solutions of x mod is,,,,,,. EXAMPLE Solve the equation x mod. Because the modulus is, try,,,,,,,, and : Is it true that mod? No Is it true that mod? No Continue trying numbers. You should find that none work except x : mod. The set of all positive solutions to the equation x mod is,,,, or,,,,,,. EXAMPLE Solve the equation x mod. Try the numbers,,,,,,, and. You should find that none work. Therefore, the equation x mod has no solutions at all. Write the set of all solutions as the empty set,. This result is reasonable since x will always be even, no matter which whole number is used for x. Since x is even and is odd, the difference x will be odd, and therefore not divisible by. EXAMPLE Solve x mod. Trying the integers,,,,,,, and shows that any replacement will work. The solution set is,,,,.

8 . Clock Arithmetic and Modular Systems An equation such as x mod in Example that is true for all values of the variable (x, y, and so on) is called an identity. Other examples of identities (in ordinary algebra) include x x x and y y y. Some problems can be solved by writing down two or more modular equations and finding their common solutions. The next example illustrates the process. EXAMPLE Julio wants to arrange his CD collection in equal size stacks, but after trying stacks of, stacks of, and stacks of, he finds that there is always disc left over. Assuming Julio owns more than one CD, what is the least possible number of discs in his collection? The given information leads to three modular equations, x mod, x mod, x mod, whose sets of positive solutions are, respectively,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, and,,,,,,,,,,,. The smallest common solution greater than is, so the least possible number of discs in the collection is. EXAMPLE A dry-wall contractor is ordering materials to finish a -footby--foot room. The wallboard panels come in -foot widths. Show that, after uncut panels are applied, all four walls will require additional partial strips of the same width. The width of any partial strip needed will be the remainder when the wall length is divided by (the panel width). In terms of congruence, we must show that mod. By the criterion given above, we see that this is true since both and give the same remainder (namely ) when divided by. A -foot partial strip will be required for each wall. (In this case four -foot strips can be cut from a single panel, so there will be no waste.) FOR FURTHER THOUGHT A Card Trick Many card tricks that have been around for years are really not tricks at all, but are based on mathematical properties that allow anyone to do them with no special conjuring abilities. One of them is based on mod arithmetic. In this trick, suits play no role. Each card has a numerical value: for ace, for two,, for jack, for queen, and for king. The deck is shuffled and given to a spectator, who is instructed to place the deck of cards face up on a table, and is told to follow the procedure described: A card is laid down with its face up. (We shall call it the starter card.) The starter card will be at the bottom of a pile. In order to form a pile, note the value of the starter card, and then add cards on top of it while counting up to. For example, if the starter card is a six, pile up seven cards on top of it. If it is a jack, add two cards to it, and so on. (continued)

9 CHAPTER Number Theory When the first pile is completed, it is picked up and placed face down. The next card becomes the starter card for the next pile, and the process is repeated. This continues until all cards are used or until there are not enough cards to complete the last pile. Any cards that are left over are put aside for later use. We shall refer to these as leftovers. The performer then requests that a spectator choose three piles at random. The remaining piles are added to the leftovers. The spectator is then instructed to turn over any two top cards from the piles. The performer is then able to determine the value of the third top card. The secret to the trick is that the performer adds the values of the two top cards that were turned over, and then adds to this sum. The performer then counts off this number of cards from the leftovers. The number of cards remaining in the leftovers is the value of the remaining top card! For Group Discussion. Obtain a deck of playing cards and perform the trick as described above. (As with many activities, you ll find that doing it is simpler than describing it.) Does it work?. Explain why this procedure works. (If you want to see how someone else explained it, using modulo arithmetic, see An Old Card Trick Revisited, by Barry C. Felps, in the December issue of the journal The Mathematics Teacher.)

### 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.

### Ready, Set, Go! Math Games for Serious Minds

Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

### 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

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### 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

### Current California Math Standards Balanced Equations

Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

### 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

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

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

### No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

### 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

### 25 Integers: Addition and Subtraction

25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers

### 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

### 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

### 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

### 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

### 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

### 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

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### Stupid Divisibility Tricks

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

### 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

### 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

### 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

### Fractions Packet. Contents

Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys..

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Unit 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 L-34) is a summary BLM for the material

### Properties of Real Numbers

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

### 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

### Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

### Math 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,

### Zero-knowledge games. Christmas Lectures 2008

Security is very important on the internet. You often need to prove to another person that you know something but without letting them know what the information actually is (because they could just copy

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### 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

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### 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

### Quotient Rings and Field Extensions

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

### 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,

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

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

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

### Permutation Groups. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003

Permutation Groups Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles April 2, 2003 Abstract This paper describes permutations (rearrangements of objects): how to combine them, and how

### 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

### RSA Encryption. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003

RSA Encryption Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles October 10, 2003 1 Public Key Cryptography One of the biggest problems in cryptography is the distribution of keys.

### 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

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### 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

### Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in

CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division

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

### The theory of the six stages of learning with integers (Published in Mathematics in Schools, Volume 29, Number 2, March 2000) Stage 1

The theory of the six stages of learning with integers (Published in Mathematics in Schools, Volume 29, Number 2, March 2000) Stage 1 Free interaction In the case of the study of integers, this first stage

### 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

### 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

### 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

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

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

### 8 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

### Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.

3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.

### Prime Time: Homework Examples from ACE

Prime Time: Homework Examples from ACE Investigation 1: Building on Factors and Multiples, ACE #8, 28 Investigation 2: Common Multiples and Common Factors, ACE #11, 16, 17, 28 Investigation 3: Factorizations:

### Mathematical Induction

Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

### Paper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 5 7

Ma KEY STAGE 3 Mathematics test TIER 5 7 Paper 1 Calculator not allowed First name Last name School 2009 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You

### Kenken For Teachers. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010. Abstract

Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 7, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills.

### CONTENTS 1. Peter Kahn. Spring 2007

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

### Adding and Subtracting Integers Unit. Grade 7 Math. 5 Days. Tools: Algebra Tiles. Four-Pan Algebra Balance. Playing Cards

Adding and Subtracting Integers Unit Grade 7 Math 5 Days Tools: Algebra Tiles Four-Pan Algebra Balance Playing Cards By Dawn Meginley 1 Objectives and Standards Objectives: Students will be able to add

### 7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School

7 th Grade Integer Arithmetic 7-Day Unit Plan by Brian M. Fischer Lackawanna Middle/High School Page 1 of 20 Table of Contents Unit Objectives........ 3 NCTM Standards.... 3 NYS Standards....3 Resources

### Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

### 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

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

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

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

### Heron Triangles. by Kathy Temple. Arizona Teacher Institute. Math Project Thesis

Heron Triangles by Kathy Temple Arizona Teacher Institute Math Project Thesis In partial fulfillment of the M.S. Degree in Middle School Mathematics Teaching Leadership Department of Mathematics University

### What Is Singapore Math?

What Is Singapore Math? You may be wondering what Singapore Math is all about, and with good reason. This is a totally new kind of math for you and your child. What you may not know is that Singapore has

### 1. 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

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### = 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

### Worldwide Casino Consulting Inc.

Card Count Exercises George Joseph The first step in the study of card counting is the recognition of those groups of cards known as Plus, Minus & Zero. It is important to understand that the House has

### Math Games For Skills and Concepts

Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### 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

### Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours

Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A3d,e Time allotted for this Lesson: 4 Hours Essential Question: LESSON 4 FINITE ARITHMETIC SERIES AND RELATIONSHIP TO QUADRATIC

### 15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012

15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012 1 A Take-Away Game Consider the following game: there are 21 chips on the table.

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

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

### 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

### 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,

### Fundamentals of Probability

Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

### The 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 15-year-old why

### LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton

LAMC Beginners Circle: Parity of a Permutation Problems from Handout by Oleg Gleizer Solutions by James Newton 1. Take a two-digit number and write it down three times to form a six-digit number. For example,

### 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

### Brain Game. 3.4 Solving and Graphing Inequalities HOW TO PLAY PRACTICE. Name Date Class Period. MATERIALS game cards

Name Date Class Period Brain Game 3.4 Solving and Graphing Inequalities MATERIALS game cards HOW TO PLAY Work with another student. Shuffle the cards you receive from your teacher. Then put them face down

### 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

### 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

### INTERSECTION MATH And more! James Tanton

INTERSECTION MATH And more! James Tanton www.jamestanton.com The following represents a sample activity based on the December 2006 newsletter of the St. Mark s Institute of Mathematics (www.stmarksschool.org/math).

### 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

### 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works.

MATH 13150: Freshman Seminar Unit 18 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works. 1.1. Bob and Alice. Suppose that Alice wants to send a message to Bob over the internet

### Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

### 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

### OA3-10 Patterns in Addition Tables

OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

### Computing exponents modulo a number: Repeated squaring

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

### Welcome to Basic Math Skills!

Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots