Number Theory. Applications of Congruences. Francis Joseph Campena Mathematics Department De La Salle University-Manila
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1 Number Theory Applications of Congruences Francis Joseph Campena Mathematics Department De La Salle University-Manila
2 Introduction Divisibility Test In this section, we discuss some divisibility rules for 10, 5, 2 i, 3,9, and 11. First, let us consider the decimal representation of a number N = n k 10 k + n k 1 10 k n n n 0.
3 Theorem (Divisibility for 10) An integer n is divisible by 10 if and only if n n 0 is the units digit of n is 0. mod 10, that Theorem (Divisibility for 5) An integer n is divisible by 5 if and only if 5 n 0, that is the units digit of n is either 0 or 5. Theorem (Divisibility for 2 i ) An integer n is divisible by 2 i if and only if the number formed by the last i digits in n is divisible by 2 i.
4 Theorem (Divisibility for 3) An integer n is divisible by 3 if and only if the k i=0 n i is divisible by 3; that is the sum of the digits of the number is divisible by 3. Theorem (Divisibility for 9) An integer n is divisible by 9 if and only if the k i=0 n i is divisible by 9; that is the sum of the digits of the number is divisible by 9. Theorem (Divisibility for 11) An integer n is divisible by 11 if and only if the even n i odd n i is divisible by 11; that is the sum of the digits of the number is divisible by 11.
5 Introduction Divisibility Test Modular arithmetic can be used to create beautiful designs. We will now explore three such designs: an m-pointed star, an (m, n) residue design, and quilt designs. They are really fun, so enjoy them.
6 m-pointed star Divisibility Test To construct an m-pointed star, mark m equally spaced points on a large circle, and label them with the least residues 0 through (m 1) modulo m. Choose a least residue i modulo m, where (i, m) = 1. Join each point x with the point x + i modulo m. Now color in the various regions inside the circle with some solid colors. You should get a nice m-pointed star.
7 (m, n)residuedesign To construct an (m, n) residue design, where 1 n m and (m, n) = 1, select m 1 equally spaced points on a large circle, label them 1 through m 1, and join each point x to point nx modulo m. Then color in the various regions formed in a systematic way to create exciting designs.
8 (m, n)residuedesign
9 (m, n)residuedesign
10 (m, n)residuedesign
11 Quilt Designs Divisibility Test We can use addition and multiplication tables for least residues modulo m to generate other artistic and interesting designs. For example, choose m = 9. Construct the addition table for the set of least residues 0 through 8 modulo 9, as shown below.
12 Quilt Designs Divisibility Test Devise nine basic design elements to represent each of the numbers 0 through 8. For instance,
13 Quilt Designs Divisibility Test Replacing each entry in the multiplication table for the least residue modulo 9 with the devised basic design, we have
14 Quilt Designs Divisibility Test For example, flip this design about its right-side edge and then flip the ensuing design about the bottom edge.
15 Details Divisibility Test 1. Form a group of 3 members. 2. Select one project from the given list. 3. Construct a module that allows you to discuss the project to a senior highschool class. 4. Provide the necessary resource guides/instructors manual, references and examples so that any teacher with sufficient background could implement the lesson.
16 Project 1: Check Digits Discuss how the theory of congruences is applied to coding theory. Specifically on the UPC (universal product codes) found on grocery items and ISBN (International Standard Book Number). You may also include other applications such as the Vehicle Identification Number of automobiles or International Article Numberting Assocication (European Article Numbering Association). Provide some examples in each application.
17 Project 2: Round Robin Tournaments Discuss how the theory of congruences is used in constructing a schedule in which each contestant meets each other contestant in turn. Provide a simple algorithm that discuss a way to construct a round robin schedule for n teams/players. Provide some examples for some small number of players/teams.
18 Project 3: Perpetual Calendar First provide a historical background on the development of the calendar that is now widely used today. Develop an alorithm to determine the day of the week for any date in any year. Provide some illustrations.
19 Project 4: Applications to Cryptography Provide a simple background on what cryptography is all about and how is the theory of congruences applied to some topics in cryptography/cryptosystems. Provide some simple illustrations.
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