Chapter 9b Applications of Modular Arithmetic PDF

CHAPTER 9. Mathematical System– Applications of Modular Arithmetic learning objectives 1. discuss variety of applications of modular arithmetic; and 2. solve problems involving modular arithmetic.  THE ISBN International Standard Book Number (ISBN) E...

CHAPTER 9. Mathematical System– Applications of Modular Arithmetic learning objectives 1. discuss variety of applications of modular arithmetic; and 2. solve problems involving modular arithmetic.  THE ISBN International Standard Book Number (ISBN) Every book that is cataloged in the National Library has an International Standard Book Number (ISBN). It is consisting of 13 digits. 𝒅𝟏 , 𝒅𝟐 , 𝒅𝟑, - first digit – composed of 978 𝒅𝟒 - next digit – indicates the country in which the publisher is incorporated (either 0 or 1) - next two to seven digits – indicate publisher … - next group of digits – indicates the title of the book 𝒅𝟏𝟑 - last digit (13th digit) – is called check digit THE ISBN Formula for the ISBN Check Digit 𝒅𝟏𝟑 = 𝟏𝟎 − 𝒅𝟏 + 𝟑𝒅𝟐 + 𝒅𝟑 + 𝟑𝒅𝟒 + 𝒅𝟓 + 𝟑𝒅𝟔 + 𝒅𝟕 + 𝟑𝒅𝟖 + 𝒅𝟗 + 𝟑𝒅𝟏𝟎 + 𝒅𝟏𝟏 + 𝟑𝒅𝟏𝟐 𝐦𝐨𝐝 𝟏𝟎 If 𝒅𝟏𝟑 = 𝟏𝟎 , then the check digit is 0. It is this check digit that is used to ensure accuracy. Example: The ISBN for the fourth edition of the American Heritage Dictionary is 978-0- 395-82517-4. Suppose, how-ever, that a bookstore clerk sends an order for the American Heritage Dictionary and inadvertently enters the number 978-0-395- 28517-4, where the clerk transposed the 8 and 2 in the five numbers that identify the book. Correct ISBN: 978-0-395-82517-4 Incorrect ISBN: 978-0-395-28517-4 THE ISBN Formula for the ISBN Check Digit 𝒅𝟏𝟑 = 𝟏𝟎 − 𝒅𝟏 + 𝟑𝒅𝟐 + 𝒅𝟑 + 𝟑𝒅𝟒 + 𝒅𝟓 + 𝟑𝒅𝟔 + 𝒅𝟕 + 𝟑𝒅𝟖 + 𝒅𝟗 + 𝟑𝒅𝟏𝟎 + 𝒅𝟏𝟏 + 𝟑𝒅𝟏𝟐 𝐦𝐨𝐝 𝟏𝟎 If 𝒅𝟏𝟑 = 𝟏𝟎 , then the check digit is 0. It is this check digit that is used to ensure accuracy. Example: Determine the ISBN check digit for the book The Equation that Couldn't Be Solved by Mario Livio. The first 12 digits of the ISBN are 978-0-7432-5820-?. THE ISBN Formula for the ISBN Check Digit 𝒅𝟏𝟑 = 𝟏𝟎 − 𝒅𝟏 + 𝟑𝒅𝟐 + 𝒅𝟑 + 𝟑𝒅𝟒 + 𝒅𝟓 + 𝟑𝒅𝟔 + 𝒅𝟕 + 𝟑𝒅𝟖 + 𝒅𝟗 + 𝟑𝒅𝟏𝟎 + 𝒅𝟏𝟏 + 𝟑𝒅𝟏𝟐 𝐦𝐨𝐝 𝟏𝟎 If 𝒅𝟏𝟑 = 𝟏𝟎 , then the check digit is 0. It is this check digit that is used to ensure accuracy. Example: A purchase order for the book The Mathematical Tourist by Ivars Peterson includes the ISBN 978-0-760-73261-6. Determine whether this is a valid ISBN. UNIVERSAL PRODUCT CODE (UPC) UPC (Universal Product Code) is a number placed on many items and is particularly useful in grocery. UPC gives the store manager accurate information about inventory and the buying habits of the store’s customers. The UPC is a 12-digit number that satisfies a congruence equation that is similar to the one for ISBNs. UNIVERSAL PRODUCT CODE (UPC) Formula for the UPC Check Digit 𝒅𝟏𝟐 = 𝟏𝟎 − 𝟑𝒅𝟏 + 𝒅𝟐 + 𝟑𝒅𝟑 + 𝒅𝟒 + 𝟑𝒅𝟓 + 𝒅𝟔 + 𝟑𝒅𝟕 + 𝒅𝟖 + 𝟑𝒅𝟗 + 𝒅𝟏𝟎 + 𝟑𝒅𝟏𝟏 𝐦𝐨𝐝 𝟏𝟎 If 𝒅𝟏𝟐 = 𝟏𝟎 , then the check digit is 0. Example: 1. Find the check digit for the DVD release of the film Alice in Wonderland if the first 11 digits are 7-86936-79798-? 2. Is 1-32342-65933-9 a valid UPC? UNIVERSAL PRODUCT CODE (UPC) Formula for the UPC Check Digit 𝒅𝟏𝟐 = 𝟏𝟎 − 𝟑𝒅𝟏 + 𝒅𝟐 + 𝟑𝒅𝟑 + 𝒅𝟒 + 𝟑𝒅𝟓 + 𝒅𝟔 + 𝟑𝒅𝟕 + 𝒅𝟖 + 𝟑𝒅𝟗 + 𝒅𝟏𝟎 + 𝟑𝒅𝟏𝟏 𝐦𝐨𝐝 𝟏𝟎 If 𝒅𝟏𝟐 = 𝟏𝟎 , then the check digit is 0. Exercises: Determine the correct digit for the UPC. a. Scrabble 6-53569-39973-? b. Beethoven’s 9th Symphony, DVD 0-32031-13439-? a. T1-84 Silver Edition Calculator 0-33317-20083-? CREDIT CARD NUMBERS Companies that issue credit cards also use modular arithmetic to determine whether a credit card number is valid. The primary coding method is based on the Luhn algorithm, which uses mod 10 arithmetic. Credit card numbers are normally 13 to 16 digits long o The first one to four digits are used to identify the card issuer. LUHN ALGORITHM It is used to determine whether a credit card number is valid and is calculated as follows: 1. Beginning with the next-to-last digit (the last digit is the check digit) and reading from right to left, double every other digit. 2. If a digit becomes a 2-digit number after being doubled, treat the number as two individual digits. 3. Find the sum of the new list of digits; the final sum must be congruent to 0 mod 10. Example: Determine whether 5234 8213 3410 1298 is a valid credit card number. LUHN ALGORITHM It is used to determine whether a credit card number is valid and is calculated as follows: 1. Beginning with the next-to-last digit (the last digit is the check digit) and reading from right to left, double every other digit. 2. If a digit becomes a 2-digit number after being doubled, treat the number as two individual digits. 3. Find the sum of the new list of digits; the final sum must be congruent to 0 mod 10. Example: Determine the correct check digit for the given Credit Card Number. 6011-0123-9145-231? LUHN ALGORITHM It is used to determine whether a credit card number is valid and is calculated as follows: 1. Beginning with the next-to-last digit (the last digit is the check digit) and reading from right to left, double every other digit. 2. If a digit becomes a 2-digit number after being doubled, treat the number as two individual digits. 3. Find the sum of the new list of digits; the final sum must be congruent to 0 mod 10. Example: Determine the correct check digit for the given Credit Card Number. 4896-4198-8760-197? End of Discussion…