# Not for resale. 4.1 Divisibility of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples

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1 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

2 4.4 Codes and Credit Card Numbers: Connections to Number Theory 269 n = 2 # 5 # Let (a) Could 2, 5, or 7 divide n evenly? Why or why not? (b) Is n prime or composite? 37. Let (a) Do any of 2, 3, 5, 7, 11, or 13 divide n evenly? Explain briefly. (b) Must n have a prime divisor different from 2, 3, 5, 7, 11, or 13? Explain briefly. n = 2 # 3 # # 11 # 13. (c) Is n prime or composite? (Hint: Use the program FACTOR on a graphing calculator.) 38. Let n = 2 # 3 # # 11 # # 19 # 23. (a) Do any of 2, 3, 5, 7, 11, 13, 17, 19, or 23 divide n evenly? Explain briefly. (b) Can you argue as in problem 37(b) that n must have a prime divisor different from 2, 3, 5, 7, 11, 13, 17, 19, and 23? (c) Find the prime-power representation of n, using the program FACTOR on a graphing calculator. 4.4 Codes and Credit Card Numbers: Connections to Number Theory Secret and not-so-secret codes have been used for centuries for both serious and recreational purposes. Aside from their considerable utility, the aura of mystery often associated with codes makes them attractive enrichment topics that constantly surprise and fascinate students. We discuss zip codes and credit card numbers here not only because they are attractive to students but also because they provide an interesting connection to number theory. International Standard Book Number (ISBN) codes will be considered in the problem set at the end of this section. Bar Codes on Envelopes and Postcards: The Postnet Code We begin with a question: What is the purpose of the strings of long and short vertical lines or bars that appear today on virtually every piece of mail? The business reply postcards shown in Figure 4.4 are typical of what we see inserted in most magazines. Surely FIGURE 4.4 Bar codes on postcards

3 270 CHAPTER 4 Number Theory the markings are placed on the cards and envelopes for a purpose. They must mean something; they must convey some sort of information. Let s take a more careful look at the bar codes shown on the two postcards in Figure 4.4. (You may also like to gather some other postcards to examine as well.) Since we suspect that the bar code is a machine-readable version of the ZIP+4 code that also appears on the cards, let s fix our attention on the nine-digit zip code numbers and their associated bar codes. The two examples are shown below in an enlarged view Looking for patterns and trying to associate the numbers with the bar code, we note that each string begins and ends with a long bar. In fact, this is always true, as you can easily check by finding other examples in your mailbox. These two bars probably tell the machine when the code starts and stops. Doing a count, we find that there are 52 bars in each code. Thus, deleting the start and stop bars, there are 50 bars to determine the zip code. Since there are nine numbers and a dash in the zip code, this may mean that all code groups are of length five. Thus, we break the bar code up into five-bar groups as shown below and try to associate the successive numbers in the zip code with the successive groups. As a check, we can use the fact that several digits are repeated both within and between the two codes. For the first code, we obtain the pairings shown. But this doesn t make sense because then 0 and the dash correspond to the same code group Indeed, given the four 0s in this code, it appears that no code group corresponds to the dash and that corresponds to 0. This means that the last two code groups both correspond to 1s, so there must be additional information in the code. Let s look again at both codes, this time leaving out the dash. We obtain these pairings Note that each of 0, 1, and 7 appears more than once in these codes and that each time they appear, they have the same representation in the bar code! This is very strong evidence

4 4.4 Codes and Credit Card Numbers: Connections to Number Theory 271 that we are correct, that the bar code represents the zip code, and that the code representations are as shown here. Of course, the last 1 and 6 in the above diagrams were determined by the earlier appearance of the same five-bar patterns in each zip code. Finally, it is not yet clear what code group should correspond to 5 and 8, but the correspondence shown below follows the pattern given by the other known digits. These code groups can be verified by looking at other pieces of mail whose zip codes contain a 5 and an We first guessed that the bar code must indeed be a code and that it likely gives the zip code in machine-readable form. We then noticed that both codes have 52 bars, beginning and ending with a long bar. Ignoring the first and last bars, we had a bar code with 50 bars. Our guess was that each five consecutive bars represented a digit that, in turn, could be determined by looking at the printed zip code. So far, we have discovered that the nine ZIP+4 digits appear in the bar code. But what is the purpose of the tenth digit that appears only in the bar code? The simplest idea is to add the ten coded digits. We find that and = = 50. It seems that the tenth digit in each case, here 1 and 6, respectively, has been chosen so that the sum of all the digits is a multiple of 10. This suggests that the tenth digit is a check digit that is included so that incorrect codes can be detected by machine. In fact, this is the case, and if an error is detected in this way, the piece of mail is automatically shunted aside for human inspection and correction of the zip code. Most pieces of mail that you receive currently have an extension of the ZIP+4 code known as the delivery point bar code. This is a twelve-digit code that begins with the ninedigit ZIP+4 code, then codes the last two digits of the address, and finally ends with the check digit that brings the sum of all twelve digits up to a multiple of 10, just as before. For example, a piece of mail delivered to 1925 SW High Street with the ZIP+4 code would use the twelve digit Postnet bar code The check digit is 9 since this makes the sum of all twelve digits = 60, a number divisible by 10. EXAMPLE 4.19 Reading Information from Postnet Bar Codes What information is encoded in each of the following Postnet bar codes? In each case, verify that the check digit is correct. (a) (b)

6 4.4 Codes and Credit Card Numbers: Connections to Number Theory 273 Step 3. Step 4. Determine the sum of the digits (not the numbers themselves) of the numbers in step 2. Determine the sum of the digits not underlined (that is, those in the oddnumbered positions) in step 1. Step 5. Add the two numbers obtained in steps 3 and 4. The number 65 is a check sum that can be computed almost instantaneously by computer. It gives a quick check (see below) that an error has or has not been made. Immediately informing the card user that an error has been made enables the user to resubmit the correct number and continue the transaction. A card number with a valid check sum is then processed further to ensure that it corresponds to an active account. But we still have not explained how the check sum reveals that an error has been made. To discover this, note that the following are all possibilities for correct credit card numbers. (a) (b) (c) = = = Computing the check sum for these three numbers (try it!), we obtain 70, 60, and 30, respectively. If you have a credit card, compute the check sum for your card and compare it with the three check sums just computed. A glance reveals that all the check sums are multiples of 10, a requirement that must be satisfied if the number is a correct possibility for a valid credit card number. EXAMPLE 4.21 Computing a Missing Credit Card Number Digit What digit must replace d to make the following a possibility for a valid credit card number? d Solution Counting from the right, we find that the digits in the even-numbered positions are 0, 6, 5, 3, 0, 3, 1, and 6, and their respective doubles are 0, 12, 10, 6, 0, 6, 2, and 12. The sum of the digits in these numbers is 21, and the sum of the digits in the odd-numbered positions is d = d + 9. Thus, the check sum is d + 30, so d must be 0 if this sum is to be a multiple of 10. Problem Set 4.4 Understanding Concepts 1. Determine the check digit for each of these ZIP+4 codes. (a) (b) (c) Write the bar code symbol for the check digit of each of these ZIP+4 codes. (a) (b) (c)

8 4.4 Codes and Credit Card Numbers: Connections to Number Theory 275 Thinking Cooperatively For the following problems, work with two or three other students. Your answers should be consensus answers for the entire group. 15. Suppose the last four digits of your credit card are Discuss what you think may happen if you make a transposition error when typing in the last four digits of your credit card number to make an on-line purchase. What happens if you mistakenly type (a) 7409? (b) 4079? (c) 4790? 16. When people submit credit card numbers on-line, do you think most, or just a few, transposition errors will be quickly detected? Experiment with different numbers to gather data. Is there something that credit card companies might do to detect all transposition errors? Making Connections 17. Between the years 1970 and 2006, all books published anywhere in the world were given an identifying number called an International Standard Book Number (ISBN). These numbers facilitated buying and selling books, inventory control, and so on. A typical ISBN is and, somewhat like a zip code, the last digit is a check digit. It works like this: There are ten digits in the code, and the check digit is chosen so that 10 times the first digit plus 9 times the second digit, plus 8 times the third digit,..., plus 1 times the check digit is divisible by 11. Thus, for the ISBN above, (10 * 0) + (9 * 1) + (8 * 3) + (7 * 2) + (6 * 5) + (5 * 7) + (4 * 5) + (3 * 0) + (2 * 2) + (1 * 7) = 143, and 11 divides 143. The check digit may be an X, denoting a 10. (a) Which of these ISBNs is correct? Explain. (i) (ii) (b) Supply the check digit to complete each of these correct ISBNs. (i) (ii) (iii) Beginning January 1, 2007, book numbers went from the 10-digit code described in problem 17 to a 13-digit code. A previously published book retains the first 9 digits of its 10-digit code, and adds the prefix 978 at the front. The final 13th digit is determined by multiplying the first 12 digits alternately by 1s and 3s, adding these numbers, and finally the check digit to bring the sum up to a multiple of 10. For example, the book with a 10-digit ISBN number will be renumbered as The check digit is 3 since this makes the weighted sum 9 + 3(7) (0) (2) (2) (6) (6) + 3 = 110 a multiple of 10. (a) What 13-digit book number corresponds to the German book ? (b) What 13-digit book number corresponds to the Spanish book ? (c) A book with ISBN: will get the new code If the ISBN is mistyped as , will the error be detected? If the 13-digit book number is mistyped as , will the error be detected? (d) What number pairs ab in a 13-digit book number can be transposed without detection? That is, if ba replaces ab, when will the check sum remain divisible by 10? Use algebraic reasoning. Using a Computer 19. The following commands set up a spreadsheet to check a ten-digit ISBN code: (i) Put 10 in A1. (ii) Put =A1-1 in B1 and COPY and PASTE, or FILL, right to J1. (iii) Put the digits of the ISBN code in A2 through J2. If X appears in the code, replace it by 10. (iv) Put = A1*A2 in A3 and COPY and PASTE, or FILL, right to J3. (v) Put = SUM(A3:J3) in K3. (vi) Put = MOD(K3, 11) in L3. This places in L3 the remainder when the number in K3 is divided by 11. If L3 contains 0, the code is correct. Set up such a spreadsheet and use it to determine whether the following codes are correct. Note that once a sheet is set up, all you need do to immediately see if a given code is correct is enter the digits of the code in A2 through J2. (a) (b)

10 Chapter 4 Summary 277 Odd Perfect Numbers The ancient Greeks called a number perfect if it equals the sum of its proper divisors. For example, 6 = , 28 = , and 496 = are the first three perfect numbers. Indeed, it is known there are infinitely many even perfect numbers. Are there any odd perfect numbers? (If one exists, then it is known that it s larger than ) In this chapter, we have introduced the basic number-theoretic notions of divisibility, factoring, factors and multiples, primes and composite numbers, least common multiples, and greatest common divisors. These ideas not only are useful in other parts of mathematics and in disciplines like computer science but also provide interesting and stimulating motivational material for the elementary mathematics classroom. It is increasingly the case that number-theoretic notions appear in elementary school texts and that these ideas must be thoroughly understood by teachers. Chapter 4 Summary Key Concepts In what follows, unless expressly stated to the contrary, all symbols represent whole numbers. Section 4.1 Section 4.2 Divisibility of Natural Numbers Divisibility. If b Z 0 and there exists a number q such that a = bq, then we say that b divides a. Factor, or divisor. If b divides a, b is called a factor, or divisor, of a. Proper divisor. A proper divisor is a divisor of a number that is less than the number. Multiple. If b divides a, a is called a multiple of b. Prime. A prime is a natural number with only two factors, itself and 1. Composite number. A composite number is a natural number with more than two factors. Alternatively, c is composite if c = ab, where a and b are natural numbers and neither a nor b equals 1. Unit. The number 1, which has only one divisor, itself. Fundamental theorem of arithmetic. Every natural number greater than 1 can be represented as a product of primes in just one way, apart from order. Alternatively, every natural number greater than 1 can be represented in just one way, apart from order, as a product of powers of primes. All divisors of a number. All divisors of a number may be determined from the number s prime-power representation. Infinitude of the set of primes. There are infinitely many primes. Determining primality. Every natural number greater than 1 is either a prime or has a prime factor less than or equal to its own square root. The sieve of Eratosthenes. A method for determining all primes up to any given limit. Tests for Divisibility Divisibility of sums and differences. Let n, a, and b be natural numbers. If n divides both a and b, the n also divides their sum a + b and their difference a - b. Divisibility by 2. n is divisible by 2 provided that its units digit is 0, 2, 4, 6, or 8. Divisibility by 5. n is divisible by 5 provided that its units digit is 0 or 5. Divisibility by 10. n is divisible by 10 provided that its units digit is 0.

11 278 CHAPTER 4 Number Theory Section 4.3 Section 4.4 Divisibility by 3 or 9. n is divisible by 3 provided that the sum of the digits in its decimal representation is divisible by 3. n is divisible by 9 provided that the sum of its digits is divisible by 9. Divisibility by 11. n is divisible by 11 provided that the difference of the sum of the digits in the even positions in the decimal representation of n and the sum of the digits in the odd positions is divisible by 11. Divisibility by powers of 2. n is divisible by 2 r provided that the number named by the last r digits of n is divisible by 2 r. Greatest Common Divisors and Least Common Multiples Greatest common divisor. If d is the greatest of all common divisors of a and b, it is called their greatest common divisor or their greatest common factor. Greatest common divisor of a and b by intersection of sets. The greatest number in the intersection of the sets of divisors of a and b. Greatest common divisor of a and b from prime-power representations. The product of prime powers p s, where s is the lesser of the exponents (including 0) on p in the prime-power representations of a and b. The Euclidean algorithm. A method for finding the greatest common divisor of a and b, using the division algorithm repeatedly. Least common multiple. If m is the least of all common multiples of a and b, it is called their least common multiple. Least common multiple of a and b by the intersection of sets. The least number in the intersection of the sets of multiples of a and b. Greatest common multiple of a and b from prime-power representations. The product of prime powers p t, where t is the greater of the exponents on p in the prime-power representations of a and b. Product of the greatest common divisor and least common multiple of a and b. This product equals ab, so that ab = GCD(a, b) # LCM(a, b). Codes and Credit Card Numbers: Connections to Number Theory Bar codes on mail. Zip codes in machine-readable form. Verification of valid credit card numbers. Schemes to check correctness of credit card numbers. Vocabulary and Notation Section 4.1 Factor, divisor Proper divisor Perfect number Multiple Prime number Composite number Unit Factor tree Fundamental theorem of arithmetic Prime-power representation of n. Section 4.2 Divisibility by 2, 5, and 10 Divisibility by 4, 8, and other powers of 2 Divisibility by 3 and 9 Divisibility by 11 Simultaneous test for divisibility by 7, 11, and 13 Divisibility by powers of 2 Section 4.3 GCD(m, n), GCF(m, n) LCM(m, n) Euclidean algorithm Section 4.4 Check sums

12 Chapter Review Exercises 279 Chapter Review Exercises Section Draw rectangular diagrams to illustrate the factorings of 15, taking order into account; that is, think of as different from 2. Construct a factor tree for (a) Determine the set of all divisors of 60. (b) Determine the set D 72 of all divisors of 72. (c) Use D 60 D 72 to determine GCD(60, 72). 4. (a) Determine the prime-power representation of (b) Determine the prime-power representation of (c) Use parts (a) and (b) to determine GCD(1200, 2940) and LCM(1200, 2940). 5. Use information from the sieve of Eratosthenes in Figure 4.3 to determine whether 847 is prime or composite. 6. (a) Determine a composite natural number n with a prime factor greater than 1n. (b) Does the n in part (a) have a prime divisor less than 1n? If so, what is it?. 7. Determine natural numbers r, s, and m such that r divides m and s divides m, but rs does not divide m. 8. Use the number to determine a prime different from 3, 5, 7, 11, 13, or 17. Section # 1. 1 # 15 D 60 n = 3 # 5 # # 13 # Using mental methods, test each number for divisibility by 2, 3, 5, and 11. (a) 9310 (b) 2079 (c) 5635 (d) Test each number for divisibility by 7, 11, and 13. (a) 10,197 (b) 9373 (c) 36, Use the results of problem 9 to decide which of these are true. (a) 15 divides (b) 33 divides (c) 55 divides (d) 55 divides Let m = 3 4 # 7 2. (a) How many divisors does m have? (b) List all the divisors of m. 13. Determine d so that 2,765,301,2d3 is divisible by (a) Determine the least natural number divisible by both q and m if q = 2 and m = 2 1 # 7 3 # 11 3 # 3 # 3 5 # 7 2 # (b) Determine the largest number less than the q of part (a) that divides q. Section (a) Find the greatest common divisor of 63 and 91 by the method of intersection of sets of divisors. (b) Find the least common multiple of 63 and 91 by the method of intersection of sets of multiples. (c) Demonstrate that 16. If, and t = 2 3 # 3 1 # 7 1 # 11 3, determine each of the following: (a) GCD(r, s, t) (b) LCM(r, s, t) 17. Determine each of the following using the Euclidean algorithm: (a) GCD(119790, 12100) (b) LCM(119790, 12100) 18. Seventeen-year locusts and 13-year locusts both emerged in When will these insects descendants next emerge in the same year? Section 4.4 = 63 # 91. GCD(63, 91) # LCM(63, 91) r = 2 1 # 3 2 # 5 1 # 11 3 s = 2 2 # 5 2 # 11 2, 19. Determine the check digit for each of these codes. (a) (b) ZIP Write out the bar code for each of these zip codes. (a) (b) Write out the ZIP+4 code named by each of these bar codes. Which one, if either, is incorrect? (a) (b)

13 280 CHAPTER 4 Number Theory 22. Which of the following could be valid credit card numbers? (a) (b) Using your own credit card, transpose two adjacent digits and see if the check sum detects the transposition. 23. In each of these, determine d and e so that the number is a valid credit card number possibility. Is there only one possibility in each case? (a) de (b) d e Chapter Test 1. Using mental methods, test each of these for divisibility by 2, 3, 9, 11, and 13. (a) 62,418 (b) 222,789 m = 2 3 # 5 2 # 7 1 # 11 4 n = 2 2 # 7 2 # Let and (a) Does r divide m if why not? (b) How many divisors does m have? (c) Determine GCD(m, n). (d) Determine LCM(m, n). r = 2 2 # 5 1 # 7 2 # 11 3? Why or 3. Indicate whether each of these is always true (T) or not always true (F). (a) If a divides c and b divides c, then ab divides c. (b) If r divides s and s divides t, then r divides t. (c) If a divides b and a divides c, then a divides (b + c). (d) If a does not divide b and a does not divide c, then a does not divide (b + c). 4. Determine the check digit for the following zip code: Determine whether 281 is a prime. 6. Given that the check digit for each of these zip codes is 0, classify each as correct or incorrect and tell why. (a) (b) Use the Euclidean algorithm to determine GCD(154, 553) and LCM(154, 553). 8. Use the Euclidean algorithm to determine each of the following: (a) GCD(13,534, 997,476) (b) LCM(13,534, 997,476) 9. Which of the following could be valid credit card numbers? (a) (b) (a) Make a factor tree for (b) Write the prime-power representation of (c) Name the largest natural number smaller than 8532 that divides (d) Name the smallest number larger than 8532 that is divisible by 8532.

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