5.4 EXERCISES. Find each of the following differences on the 12-hour clock. 6. closure 7. commutative 8. identity

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1 . EXERCISES Find each of the following differences on the -hour clock Complete the -hour clock multiplication table below. You can use repeated addition and the addition table (for example, ) or use mod multiplication techniques, as in Example 8, parts (d) and (e) By referring to your table in Exercise, determine which of the following properties hold for the system of -hour clock numbers with the operation of multiplication.. closure 7. commutative 8. identity A -hour clock system utilizes the set {0,,,, }, and relates to the clock face shown here. 9. Complete this -hour clock addition table hour clock

2 . Clock Arithmetic and Modular Systems 9 Which of the following properties are satisfied by the system of -hour clock numbers with the operation of addition? 0. closure. commutative. identity (If so, what is the identity element?). inverse (If so, name the inverse of each element.). Complete this -hour clock multiplication table Which of the following properties are satisfied by the system of -hour clock numbers with the operation of multiplication?. closure. commutative 7. identity (If so, what is the identity element?) In clock arithmetic, as in ordinary arithmetic, a b d is true if and only if b d a. Similarly, a b q if and only if b q a. Use the idea above and your -hour clock multiplication table of Exercise to find the following quotients on a -hour clock Is division commutative on a -hour clock? Explain.. Is there an answer for 0 on a -hour clock? Find it or explain why not. The military uses a -hour clock to avoid the problems of A.M. and P.M. For example, 00 hours is 0 A.M., while 00 hours is 9 9 P.M. ( noon 9 hours). 0 In these designations, the last two digits represent minutes, and the digits before that represent hours. Find 7 each of the following sums in the -hour clock system Explain how the following three statements can all be true. (Hint: Think of clocks.) Answer true or false for each of the following mod 0. 8 mod 9. 0 mod mod Work each of the following modular arithmetic problems.. 7mod. 9mod 9. mod. 8 mod 7. 8mod 8. mod 8 9. mod mod 0. The text described how to do arithmetic mod m when the ordinary answer comes out nonnegative. Explain what to do when the ordinary answer is negative. Work each of the following modular arithmetic problems.. 7mod. 0mod 7. 8 mod. mod In each of Exercises and 7: (a) Complete the given addition table. (b) Decide whether the closure, commutative, identity, and inverse properties are satisfied. (c) If the inverse property is satisfied, give the inverse of each number.. mod 7. mod

3 0 CHAPTER Number Theory In each of Exercises 8 : (a) Complete the given multiplication table. (b) Decide whether the closure, commutative, identity, and inverse properties are satisfied. (c) Give the inverse of each nonzero number that has an inverse. 8. mod 9. mod mod. mod Complete this statement: a modular system satisfies the inverse property for multiplication only if the modulus is a(n) number. Find all positive solutions for each of the following equations. Note any identities.. x mod 7. x 7 mod. x mod. x 7 mod Solve each problem. 7. Odometer Readings For many years automobile odometers showed five whole number digits and a digit for tenths of a mile. For those odometers showing just five whole number digits, totals are recorded according to what modulus? 8. Distance Traveled by a Car If a car s five-digit whole number odometer shows a reading of 9,0, in theory how many miles might the car have traveled? 9. Piles of Ticket Stubs Lawrence Rosenthal finds that whether he sorts his White Sox ticket stubs into piles of 0, piles of, or piles of 0, there are always left over. What is the least number of stubs he could have (assuming he has more than )? 0. Silver Spoon Collection Roxanna Parker has a collection of silver spoons from all over the world. She finds that she can arrange her spoons in sets of 7 with left over, sets of 8 with left over, or sets of with left over. If Roxanna has fewer than 00 spoons, how many are there?. Determining Day of the Week Refer to Example 9 in the text. (Recall that next year is a leap year.) Assuming today was Thursday, January, answer the following questions. (a) How many days would the next year (starting today) contain? (b) What day of the week would occur one year from today?. Determining a Range of Dates Assume again, as in Example 9, that next year is a leap year. If the next year (starting today) does not contain days, what is the range of possible dates for today?. Flight Attendant Schedules Robin Strang and Kristyn Wasag, flight attendants for two different airlines, are close friends and like to get together as often as possible. Robin flies a -day schedule (including days off), which then repeats, while Kristyn has a repeating 0-day schedule. Both of their routines include stopovers in Chicago, New Orleans, and San Francisco. The table below shows which days of each of their individual schedules they are in these cities. (Assume the first day of a cycle is day number.) If today is July and both are starting their schedules today (day ), list the days during July and August that they will be able to see each other in each of the three cities. The basis of the complex number system is the imaginary number i. The powers of i cycle through a repeating pattern of just distinct values as shown here: i 0, i i, i, i i, i, i i, and so on. Find the value of each of the following powers of i i 7 i Days in Days in Days in New San Chicago Orleans Francisco Robin,, 8,, 8, 9 Kristyn, 9, 0,, 7 8, 0,, 0, i 7 i 98

4 . Clock Arithmetic and Modular Systems The following formula can be used to find the day of the week on which a given year begins.* Here y represents the year (after 8, when our current calendar began). First calculate a y y y 00 y 00, where x represents the greatest integer less than or equal to x. (For example, 9. 9, and.) After finding a, find the smallest nonnegative integer b such that a b mod 7. Then b gives the day of January, with b 0 representing Sunday, b Monday, and so on. Find the day of the week on which January would occur in the following years Some people believe that Friday the thirteenth is unlucky. The table* below shows the months that will have a Friday the thirteenth if the first day of the year is known. A year is a leap year if it is divisible by. The only exception to this rule is that a century year (900, for example) is a leap year only when it is divisible by 00. First Day of Year Non-leap Year Leap Year Sunday Jan., Oct. Jan., April, July Monday April, July Sept., Dec. Tuesday Sept., Dec. June Wednesday June March, Nov. Thursday Feb., March, Nov. Feb., Aug. Friday August May Saturday May Oct. Use the table to determine the months that have a Friday the thirteenth for the following years Modular arithmetic can be used to create residue designs. For example, the designs (, ) and (, ) are shown at the top of the next column (, ) (, ) To see how such designs are created, construct a new design, (, ), by proceeding as follows. (a) Draw a circle and divide the circumference into 0 equal parts. Label the division points as,,,, 0. (b) Since mod, connect and. (We use as a multiplier since we are making an (, ) design.) (c) 0 mod Therefore, connect and. (d) (mod ) Connect and. (e) (mod ) Connect and. (f) (mod ) Connect and. (g) (mod ) Connect and. (h) 7 (mod ) Connect 7 and. (i) 8 (mod ) Connect 8 and. (j) 9 (mod ) Connect 9 and. (k) 0 (mod ) Connect 0 and. (l) You might want to shade some of the regions you have found to make an interesting pattern. For more information, see Residue Designs, by Phil Locke in The Mathematics Teacher, March 97, pages 0. *Given in An Aid to the Superstitious, by G. L. Ritter, S. R. Lowry, H. B. Woodruff, and T. L. Isenhour. The Mathematics Teacher, May 977, pp. 7.

5 CHAPTER Number Theory Identification numbers are used in various ways for many kinds of different products. * Books, for example, are assigned International Standard Book Numbers (ISBNs). Each ISBN is a ten-digit number. It includes a check digit, which is determined on the basis of modular arithmetic. The ISBN for one version of this book is The first digit, 0, identifies the book as being published in an English-speaking country. The next digits,, identify the publisher, while 808 identifies this particular book. The final digit, 9, is a check digit. To find this check digit, start at the left and multiply the digits of the ISBN by 0, 9, 8, 7,,,,, and, respectively. Then add these products. For this book we get The check digit is the smallest number that must be added to this result to get a multiple of. Since 9, a multiple of, the check digit is 9. (It is possible to have a check digit of 0; the letter X is used instead of 0.) When an order for this book is received, the ISBN is entered into a computer, and the check digit evaluated. If this result does not match the check digit on the order, the order will not be processed. Which of the following ISBNs have correct check digits? Find the appropriate check digit for each of the following ISBNs. 79. A Beautiful Mind, by Sylvia Nasar, Brunelleschi s Dome, by Ross King, Empire Falls, by Richard Russo, Longitudes and Attitudes, by Thomas L. Friedman,