# Chapter 11 Number Theory

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1 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 of number theory. It is therefore somewhat ironic that number theory now plays important roles in keeping military and diplomatic messages secret and in making certain that people are authorized to withdraw money in electronic financial transactions. i (These naturally are more complicated than your secret PIN for an automatic teller machine.) Our attention will be restricted largely to the ideas from number theory that come up in the elementary school curriculum. Although number theory ideas can be applied to negative integers as well as positive ones, we will have in mind only the whole numbers, 0, 1, 2, 3,..., in this section. Fractions use number theory ideas, but only in a context where number theory is applied to whole-number numerators and denominators Factors and Multiples, Primes and Composites Numbers are related to one another in many ways. In this section, we examine the fundamental ways that whole numbers exist when they are expressed multiplicatively. Recall the discussion of factors in Chapter 3. Since 5 6 = 30, and 15 2 = 30, each of the numbers 5, 6, 15 and 2 is a factor of 30. (There are more.) Some numbers have many more factors. Indeed, 180 has 18 factors in all! Even a small number can have several factors: 2 3 = 6 and 1 6 = 6, so the numbers 2, 3, 1, and 6 are factors of 6. Some numbers have exactly two different factors for example, 13 has only 1 and 13 as factors. Such numbers play an important role in number theory and are called prime numbers. The number 29 is another example of a number that has exactly two factors: 1 and 29, so 29 is a prime number. It may be surprising to you that there are infinitely many prime numbers, a fact known to the ancient Greeks. There are, for example, 455,052,512 prime numbers less than Indeed, with the advancing capabilities of computers (and knowledge of number theory) larger and larger primes are occasionally found. In 1978, for example, the largest known prime required 6533 digits to write. By 1985 other new primes had been found, the largest one requiring 65,050 digits to write. (How many pages would that require?) By 1992, mathematicians had found a prime number requiring 227,832 digits to write. In 1997, they found a prime requiring 895,932 digits, which would fill 450 pages of a paperback book. As of this writing, the largest 229

2 230 Chapter 11 Number Theory known prime number has 7,816,230 digits there is good reason for not printing it here! In fact, if you were able to write 10 digits per second (a feat in and of itself) it would take you 9 days to write this number. A prime number is a whole number that has exactly two different whole number factors. A composite number is a whole number greater than 1 that has more than two factors. THINK ABOUT... Does the number 1 fit the description of a prime number? Of a composite number? What about the number 0? Discussion 1 Representing Primes and Composites Suppose you have n tiles or counters as chairs. If n = 6, in how many different ways can you arrange the chairs in complete rows with the same number in each row? if n = 13? if n = 14? Discuss how the possible arrangements of n chairs in a rectangular array are different for n as a prime versus n as a composite number. Eratosthenes, a Greek who lived more than 2200 years ago, devised the following method of identifying primes. Activity 1 Eratosthenes s Sieve for Finding Primes 1. Cross out 1 in the array on the following page. The number 2 is prime. Circle 2 in the array. Cross out all of the larger multiples of 2 in the array (2 2 = 4, 3 2 = 6, 4 2 = 8... ). The number 3 is prime. Circle 3 in the array. Cross out all of the larger multiples of 3 in the array. The number 5 is prime. Circle 5 in the array. Cross out all of the larger multiples of 5 in the array. The number 7 is prime. Circle 7 in the array. Cross out all of the larger multiples of 7 in the array. What is circled next? Does this procedure ever end? Explain. Circle 11 in the array. Cross out all of the larger multiples of 11 in the array. Circle all of the numbers not yet crossed out. Are the numbers circled all primes?

3 Section 11.1 Factors and Multiples, Primes and Composites etc. 2. If this array were extended, which column would 1000 be in? Which column would 1,000,000 be in? 2 10 = Which column would this number be in? 2 11 = Which column would this number be in? Continue on the next page.

4 232 Chapter 11 Number Theory 3. Find a column in the array for which the following is true: If two numbers in the column are multiplied, the product is also in that column. Each of four original columns in the array (if continued infinitely) has this property. Which four? What is the name of this property? THINK ABOUT... If this array were written in four columns rather than six columns, which column would the number 1000 be in? How did you determine that? You probably observed that all the numbers in the last column of the array are multiples of 6. You can represent each of these numbers as rectangular arrays with exactly 6 tiles in each row. Rectangular arrays can be used to illustrate some relationships between numbers. You can easily draw a rectangular array of 18 tiles with 6 in each row. This rectangular array can illustrate the following: 3 6 = is a multiple of is the product of 6 and 3, which are factors of 18. THINK ABOUT... Can you draw a rectangular array with 15 tiles that has 6 tiles in each row? Why or why not? You should recall the following definitions from earlier work. Pay particular attention to these vocabulary words. They are often misused. If mn = p, then m and n are called factors of p, and p is called a multiple of m (and of n). If mn = p and m is not 0, then m is called a divisor of p. We say that p is divisible by m. Recall also that p is the product of m and of n. EXAMPLE = 180, so 12 and the 15 are factors of 180; they are also divisors of is the product of 12 and 15 or, in number theory lingo, 180 is a multiple of 12 (and of 15). 0 5 = 0, so 0 and 5 are factors of 0, and 5 is a divisor of 0 but 0 is not a divisor of 0. THINK ABOUT... Why is 0 never a divisor? Think back to Chapter 3 in which dividing by 0 was discussed.

5 Section 11.1 Factors and Multiples, Primes and Composites 233 Activity 2 Vocabulary Practice 1. Use 6, 8, and 48 in sentences that involve factor, divisor, product, and multiple. Use rectangular arrays, equations, and words to describe the relationship between 6, 8, and Use these vocabulary words to describe the variables in mn = p. 3. If m is a factor of n, is n a multiple of m? If p is a multiple of q, is q a factor of p? 4. If m = 2n, what can you say about m? The following activity will help us think about these relationships in different ways. Activity 3 Which Lockers Are Open? In a certain school there are 100 lockers lining a long hallway. All are closed. Suppose 100 students walk down the hall, in file, and the first student opens every locker. The second student comes behind the first and closes every second locker, beginning with locker #2. The third student changes the position of every third locker; if it is open this student closes it; if it is closed, this third student opens it. The fourth student changes the position of every fourth locker, and so on, until the 100th student changes only the position of the 100th locker. After this procession, which lockers are open? Why are they open? At the end of this processon, how many times did locker 9 get changed? How many times did locker 10 get changed? TAKE-AWAY MESSAGE... Understanding distinctions between prime and composite numbers, and between multiples and factors (or divisors) are essential before continuing into the remaining sections of this chapter. The Sieve of Eratosthenes provides one way to find prime numbers, but is not efficient for large numbers. Prime numbers are used in cryptography and in businesses such as banking. Learning Exercises for Section a. Give three factors of 25. Can you find more? If so, how? If not, why not? b. Give three multiples of 25. Can you find more? If so, how? If not, why not? 2. a. Write an equation that asserts that 25 is a factor of k. How could a rectangular array show this? b. Write an equation that asserts that m is a factor of w. c. Write an equation that asserts that v is a multiple of t. 3. a. If 216 is a factor of 2376, what equation must have a whole number solution? b. How does one find out whether 144 is a factor of 3456? 4. Use the notion of rectangular arrays to assert that 21 is not divisible by 5.

6 234 Chapter 11 Number Theory 5. Explain why these assertions are not quite correct: a. A factor of a number is always less than the number. b. A multiple of a number is always greater than the number. 6. a. Give two factors of 506. b. Give two multiples of True or false? If false, correct the statement. a. 13 is a factor of 39. b. 12 is a factor of 36. c. 24 is a factor of 36. d. 36 is a multiple of 12. e. 36 is a multiple of 48. f. 16 is a factor of 512. g. 2 is a multiple of a. Write an equation that asserts that 15 is a multiple of a whole number k. b. Write an equation that asserts that a whole number m is a factor of a whole number x. 9. a. Suppose that k is a factor of m and m is a factor of n. Is k a factor of n? Is n a multiple of k? Justify your decisions. b. Suppose that k is a factor of both m and n. Is k a factor of m + n also? Justify your decision. c. Suppose k is a factor of m but k is not a factor of n. Is k a factor of m + n also? Justify your decisions. (You may want to try this with numbers first. For example, 5 is a factor of 15, but is not a factor of.) 10. You know that the even (whole) numbers are the elements of the set of numbers 0, 2, 4, 6, 8,..., and that the odd (whole) numbers are the elements of the set of numbers 1, 3, 5, 7, 9,.... a. Write a description of the even numbers that uses 2 and the word factor. b. Write a description of the even numbers that uses 2 and the word multiple. c. Write a description of odd numbers that uses Complete the following addition and multiplication tables for even and odd numbers. Can you then make any definite assertions about... + even odd even odd even even odd odd a. the sum of any number of even numbers? b. the sum of any number of odd numbers? c. the product of any number of even numbers?

7 Section 11.2 Prime Factorization 235 d. the product of any number of odd numbers? e. whether it is possible for an odd number to have an even factor? f. whether it is possible for an even number to have an odd factor? g. Is the set of even numbers closed under addition? h. Is the set of odd numbers closed under addition? i. Is the set of even numbers closed under multiplication? j. Is the set of odd numbers closed under multiplication? 12. Explain why each of these is a prime number: 2, 3, 29, List all the primes (prime numbers) less than 100. (You can use the Sieve of Eratosthenes in the activity on the sieve.) 14. Explain why each of these is a composite number: 15, 27, 49, a. Why is 0 neither a prime nor a composite number? b. Why is 1 neither a prime nor a composite? c. What is the drawback to the following definition of prime numbers: a whole number with only 1 and itself as factors? 16. Give two factors of each number (there may be more than two): a. 829 b c. 506 d. n (if n > 1) 17. Explain why 2 is the only even prime number. (Can you always find a third factor for larger even numbers?) 18. Conjecture: Given two whole numbers, the larger one will have more factors than the smaller one will. Gather more evidence on this conjecture by working with several (4 or 5) pairs of numbers. 19. a. Just above a number line (at least to 50), mark each factor of 24 with a heavy dot and mark each multiple of 6 with a square. b. Just below the same number line, mark each factor of 18 with a triangle and mark each multiple of 18 with a circle. c. What are common factors of 18 and 24? What are common multiples of 6 and 18? 20. Explain without much calculation how you know that 2, 3, 5, 7, 11, 13, and 17 are not factors of n = is called a perfect number because its factors (other than itself) add up to the number: = 6. What is the next perfect number? 11.2 Prime Factorization How do you know whether a number is prime or composite? The number 6 can be written as a product of prime numbers: 2 3. The number 18 can be written as the product of three primes: Can other composite numbers be written as a product of primes? These questions and others are explored in the next activity.

8 236 Chapter 11 Number Theory Activity 4 I m in My Prime Write the numbers from 2 to 48 using only prime numbers = = Every whole number except 1 can be written as a prime number or as a product of prime numbers. (Why is that?) A number written as a product of prime numbers is in prime factorization form. Continuation of Activity 4 I m in My Prime a. Compare your table with others. b. Write down as many patterns as you can find in this table. c. If the list continued, what would be the prime factorization of 1008? d. Did you have any prime factorizations different from those of other people? e. What can you say about numbers that are divisible by 2? By 5? By 10? All of you should have exactly the same factorizations in Activity 4, except possibly for the order of the factors (and notational shortcuts like 3 2 for 3. 3). This result is true in general. The fact that every whole number greater than 1 is either a prime or can be expressed as the product of prime numbers uniquely (except possibly for order) is called the Unique Factorization Theorem, or sometimes, the Fundamental Theorem of Arithmetic.

9 Section 11.2 Prime Factorization 237 The Fundamental Theorem of Arithmetic means that, in some sense, the prime numbers can be regarded as the building blocks for all the whole numbers other than 0 and 1. Other whole numbers are primes or can be expressed in exactly one way as the product of primes. Thus a number theorist often finds it most useful to think of 288 as , or By asking that the prime factors be given in increasing order and that exponents be used if a prime factor is repeated, one gets the standard prime factorization. For example, 180 = and 288 = This form is not essential, but it makes quick comparisons of two prime factorizations easier. Activity 5 Detective Work on Factorizations 1. Why doesn't this correct equation contradict the Unique Factorization Theorem? = These people are finding the prime factorization of the same number x. No one is finished. Answer (and explain) the questions below without doing any computation: Aña: x = Ben: x = an odd number Carlos: x = an even number Dee: x = an odd number a. Who might agree when they finish? b. Who definitely will disagree? c. Might they all be correct, if they work forward from where they are now? One consequence of unique factorization into primes is that any factor (except 1) of a number greater than 1 can involve only primes that appear in the number s prime factorization. For example, suppose that n = Then 42 is a factor of 2100 because = If we continue to factor the 42 and 50, eventually we will get prime factors that must appear in the prime factorization of 2100 because the prime factorization is unique. Any prime number that does not appear in the prime factorization of 2100 cannot be hidden in some factor of the number. One way to visually organize the work when finding the prime factorization of a number is to make a factor tree. Thus:

10 238 Chapter 11 Number Theory Looking at the ends of each branch of the previous factor tree, we have factors 3, 7, 2, 2, 5, 5. Thus 2100 = Note that we could have thought of the composite numbers in other ways, for example, 42 as 6 7, or 2100 as THINK ABOUT... If different factorizations for composite numbers in the factor tree are used, will the prime factorization be the same? The general argument for finding the prime factorization of a number follows the same pattern. Suppose that m is a factor of n. Then there is some whole number k such that m. k = n. This process starts a factorization of n, and the factorization must lead eventually to a unique set of prime factors, according to the unique factorization theorem. This result means that some of the same primes in the prime factorization of n must be factors of m (and of k) and that no other primes can be involved. Activity 6 More On Primes 1. If n = , what are four prime factors of n? What are ten composite factors of n? 2. Find the smallest number with factors... a. 2, 6, 8, 22, 30, and 45. b. 50, 126, and 490. How many factors does a number have? For example, consider the number 72. We could write 72 as 6 12, or 2 36, or 3 24, or 4 18 or 8 9, or as We could organize these by listing each pair of numbers. In other words, we can find pairs of buddy factors. 1, 2, 3, 4, 6, 8, match with 72, 36, 24, 18, 12, and 9, respectively (i.e., 1 and 72 are buddy factors, 2 and 36 are buddy factors, etc.). So 72 has 12 factors. We can find the prime factorization of 72 by using a factor tree or by simply factoring 72 as 6 12, and then continuing by writing 6 and 12 as a product of prime numbers. The prime factorization of 72 is (2 3) (2 2 3), or How can we use the prime factorization of a number to list all the factors of a number? We could use the knowledge of the prime factorization of 72 = to list the factors systematically as follows: ; ; ; gives us factors 1, 2, 4, and ; ; ; gives us factors 3, 6, 12, and ; ; ; gives us factors 9, 18, 36, and 72.

11 Section 11.2 Prime Factorization 239 THINK ABOUT... What is the same about each row of factors? What is different between the rows of factors? Counting is one of the big ideas of mathematics that can be found throughout mathematics. The questions Do we have them all? and Are any repeats? can usually only be answered if the counting was systematic. Be sure to note how the system worked to assure finding all factors. Activity 7 Use My Rule Systematically list all the factors of 5000 = You should have 20 numbers listed. Understanding the system will be critical in generalizing a rule that will help you effectively predict the number of factors a number has. Discussion 2 How Many Factors? Return to your table in Activity 4 and write each number with exponents where possible. How can you determine the number of factors of a composite number by knowing the exponents in its prime factorization? Determine a rule that will tell you the number of factors for any whole number except 0. Hint: for any prime number a, to the nth power, a can appear in a factor in n + 1 ways where the exponents are... (you finish; hint, see the rows of factors of 72 above). Take note of the difference between finding all factors and finding the prime factors. TAKE-AWAY MESSAGE... Every whole number greater than 1 can be uniquely factored into primes (disregarding order). This fact is referred to as the Unique Factorization Theorem (sometimes as the Fundamental Theorem of Arithmetic). Another way to think about factors of a number is to list all factors of the number. A buddy system for finding all factors of a number was described, but a better way to ensure you have listed them all is to systematically list all factors using the prime factorization, and taking every possible selection from the prime factors. Learning Exercises for Section State the unique factorization theorem. What does it assert about 239,417? 2. Find the prime factorization of each of the following, using a factor tree for each. a. 102 b c d e. 121 f. 1485

12 240 Chapter 11 Number Theory 3. Find the prime factorization of each of these numbers, using a factor tree for at least two of them. a b. 256 c d e. 17,280 f. Does a complete factor tree for a number show all the factors of the number? All the prime factors of the number? 4. Name three prime factors of each of the following products. a b c What is the difference between prime factor and prime factorization? 6. Is it possible to find nonzero whole numbers m and n such that 11 m = 13 n? Explain. 7. Which cannot be true, for whole numbers m and n? Explain why not. For the ones that can be true, give values for m and n that make the equation true. a = m b = m c = n d = m e. 4 m = 8 n f. 6 m = 18 n 8. Consider m = Without elaborate calculation, tell which of the following could NOT be factors of m. Explain how you know. a b c d e If 35 is a factor of n, give two other factors of n (besides 1 and n). 10. How many factors does each have? a. 2 5 b = 108 c. 45,000 d e f g Explain your reasoning for two of the parts (a) (g). 11. Consider Which of the following products of given numbers are factors of this number for some whole number n? If so, provide a value of n that makes it true. If not, tell why not. a n b n c n d n 12. Consider q = Which of the following are multiples of this number? If so, what would you need to multiply this q by to get the number? a b c d. ( ) a. How many factors does 64 have? List them. b. How many factors does 48 have? List them. c. How many factors does have? 14. If p, q, and r are different primes, how many factors does each of the following have? a. p 10 b. p m c. q n d. p m. q n e. p m. q n. r s

13 Section 11.3 Divisibility Tests to Determine Whether a Number Is Prime Give two numbers that have exactly 60 factors. (The numbers do not have to be in calculated form.) 16. Give one number that has the number 121 as a factor and that also has exactly 24 factors. Is there just one possibility? 11.3 Divisibility Tests to Determine Whether a Number Is Prime The secret military and diplomatic codes mentioned earlier usually involve knowing whether a large number is a prime, or finding the prime factors of a large number. It is a challenge to tackle a large number like 431,687 to see whether it is prime. (431,687 is not a large number for a computer, however. It was newsworthy in 1995 that it was possible to find out whether a number 129 digits long was a prime, using a network of 600 volunteers with computers. It took them eight months and about calculations.) Because a large number automatically has two different factors, 1 and itself, we need find only one other factor to settle the question of whether or not the number is prime. If it has a third factor, we know that the number is not a prime. If primeness is the only concern, we do not even have to look for other factors. Activity 8 Back to Patterns in the Table In Section 11.2 of this chapter, you were asked to find patterns in the Activity 4 table. Did you find patterns that would tell you when a number is divisible by 2? By 3? By 4? By 5? By 6? See if you can find any now. Is 495,687,115 a prime? Is 1,298,543,316 a prime? It is likely that you saw immediately that 5 is a factor of the first number, and that 2 is a factor of the second number, so you quickly knew that each number had at least 3 factors and hence was not a prime. There are other divisibility tests beyond those for 2, 5, and 10. Perhaps you found some in the activity. Divisibility tests are useful in investigating whether a given number is a prime (and for a teacher who wishes to make up division problems that come out even that is, give a remainder of 0). They can be helpful when considering whether pairs of numbers have factors in common when writing equivalent fractions and, later, in factoring algebraic expressions. A divisibility test tells whether a number is a factor or divisor of a given number, but without having to divide the given number by the possible factor. So they could be called factor tests but usually are not. Nor are they called multiples tests, even though, for example, 112 is a multiple of 2 (but not of 5).

14 242 Chapter 11 Number Theory Here are possible statements for the divisibility tests for 2 and for 5. Divisibility Test for 2: A number is divisible by 2 if, and only if, 2 is a factor of the ones digit (i.e., the final digit is 0, 2, 4, 6, or 8). Divisibility Test for 5: A number is divisible by 5 if, and only if, 5 is a factor of the ones digit (i.e., the final digit is 0 or 5). THINK ABOUT... The divisibility tests for 2, 5, and 10 ignore most of the digits in a large number! Why do the tests work? There are some easy-to-use but difficult-to-explain divisibility tests. They can be demonstrated using properties of operations and using the two conjectures that you examined in Section 11.1, Learning Exercise 9: (1) if k is a factor of both m and n, then k is also a factor of m + n, and (2) if k is a factor of m but k is not a factor of n, then k is not a factor of m + n. THINK ABOUT... Test these conjectures by putting in different numbers for each of the conjectures to be sure you understand them. (For example, suppose k is 3, m is 12, and n is 15. Does the first conjecture hold true? Does k also divide m + n?) The test for divisibility by 3 is quite different from the test for divisibility by 2. The last digit of a number does not reveal whether a number is divisible by 3. For example, 26 and 36 both end in a digit which represents a number divisible by 3, but 26 is not divisible by 3, whereas 36 is. Furthermore, we cannot simply say that every third number in the table in Section 11.2 is divisible by 3 and have an efficient test for a large number. Of course, if we know the prime factorization we can tell immediately whether a number is divisible by 3, but one reason we need divisibility tests is to find the prime factorization of a number. Thus we know that = 168, so 168 is divisible by 3 because it has 3 as a factor. How could you tell that 3 is a factor of 168, except by calculating or knowing its prime factorization? That is, what is a divisibility test for 3? Looking at the expanded place-value expression for 168 tells the secret. (Note which properties are used in the equations below.) 168 = = 1. (99 + 1) + 6. (9 + 1) + 8 = (using the distributive property of multiplication over addition) = (using the commutative property of addition) = ( ) + ( ) (using the associative property of addition)

15 Section 11.3 Divisibility Tests to Determine Whether a Number Is Prime 243 The numbers 99 and 9 are always divisible by 3, so ( ) is divisible by 3 by the first conjecture which you tested in the Think About. And by the second conjecture, 3 would need to divide the rest of the number ( ) if 3 indeed divides 168. But , is just , which is the sum of the digits of 168. So if 3 divides , then 3 divides 168. Three does divide = 15, so 168 must be divisible by 3. (If we actually did the calculation to check whether 3 is a factor, we would find that 168 = 3. 56, so indeed 3 is a factor of 168.) THINK ABOUT... Does the following approach work for 3528? 3528 = = 3. ( ) + 5. (99 + 1) + 2. (9 + 1) + 8 = ( ) + ( ) You finish checking for divisibility by 3 using the process used for 168. Can you test for divisibility for 3 for any number, using this process? However, if the number had been 3527, all would be the same except the would now be = 17. The number 17 is not divisible by 3, and so 3 is not a factor of (Check it out using a calculator or long division.) Divisibility Test for 3: A whole number is divisible by 3 if, and only if, the sum of the digits of the whole number is divisible by 3. An interesting and useful fact about dividing a number by 9 is that the remainder for the division is always the sum of the digits of the number, if the digits continue to be added until the number is less than 9. For example, = 23 remainder 8 and = 8, the remainder. Another way of writing this is 215 = Why does this work? 215 = = 2(99 + 1) + 1(9 + 1) + 5 = (2 99) + (2 1) + (1 9) + (1 1) + 5 = [(2 9 11) + (1 9)] + ( ) = 9[ ] + ( ) = and note that the 8 = Activity 9 What About 9? Use the reasoning for the divisibility test for 3 and the reasoning about the remainder when dividing by 9 to devise a divisibility test for 9. Sometimes children are taught a method for checking arithmetic calculations called Casting Out Nines. Perhaps you know it. This method is easy enough for an upper elementary student to use, but understanding why it works involves some of the notions we ve been discussing. Here s how it works if you wish to check for errors in addition:

16 244 Chapter 11 Number Theory Step 1: For each number, cross out ( cast out ) digits that are 9 or whose sum is 9. Step 2: Add the remaining digits until you have a number 0 8. This will be your reduced number. Step 3: Do the operation indicated on the reduced numbers. Step 4: Check to see if the sum, difference, or product, as appropriate, of the reduced numbers matches the reduced number of the sum, difference, or product. If it doesn t, check further for an error. For the above addition problem, here is an illustration of the method: /2 / 2 (also, when 326 is divided by 9 the remainder is 2) 47 / 4+7=11; 1+1= 2 (also, when 479 is divided by 9 the remainder is 2) =12; 1+2=3 (also, when 84 is divided by 9 the remainder is 3) 88/ 8+8=16; 1+6=7 (also when 889 is divided by 9 the remainder is 7) If the sum is correct, then the sum of 2, 2, and 3 should equal the reduced number of the original sum. It does! = 7. Why does this work? It is based on the fact that the sum of the digits of a number is the remainder when dividing by 9. Consider: 326 = (note that = 11 and that = 2) 479 = (note that = 20 and that = 2) + 84 = (note that = 12 and that = 3) 889 should equal 9( ) + ( ) = 9(98) + 7, and it does. A similar technique works for checking products. Activity 10 Casting Out Nines Try this technique with another sum of three large numbers using a new set of numbers. Include a number whose digits sum to 9 and a number with 9 as a digit or two digits that sum to 9. Did the sum of the reduced numbers equal the reduced number of the sum? Note that if one digit in the sum is changed (as when an error is made), the reduced number of the sum is different. Generate a multiplication problem and use the analogous set of steps for multiplication. Now, the most important question that begs to be asked is, why does this work? Hint: What are the reduced numbers? Another important mathematical question needs to be answered: Are there circumstances when using the method would fail to catch an error? As we have seen thus far, divisibility tests can involve looking at the last digit or summing all the digits. What about a divisibility test for 4? In the Activity 4 table in Section 11.2, numbers divisible by 4 had 2 2 as a factor. Did you notice anything

17 Section 11.3 Divisibility Tests to Determine Whether a Number Is Prime 245 about the last two digits of each of these numbers? Consider again the number 3528: 3528 = = ( ) + ( ). Notice that 1000 is divisible by 4 because 1000 = , so must be divisible by 4. Similarly, is divisible by 4 because 100 = We are left with or 28. If this is divisible by 4, then the entire number 2528 must be divisible by 4. With similar reasoning, 2527 is not divisible by 4. Divisibility Test for 4: A number n is divisible by 4 if, and only if, 4 is a factor of the number formed by the final two digits of n. Activity 11 Divisible by 8? Use the reasoning from the divisibility test for 4 to construct a divisibility test for 8. (Consider the last three digits of the number.) Return again to the Activity 4 table in Section 11.2 and note which numbers are divisible by 6. Note that they all have 2 and 3 as factors. Thus, applying the divisibility rules for 2 and for 3 will show whether a number is divisible by 6. What about a divisibility test for 12? Again from the table, all of 12, 24, 36, 48 and 60 have 2. 3 as a factor, but so do 18 and 30, which are not divisible by 12. So just testing for 2 and 3 will not suffice as a divisibility test for 12. All numbers divisible by 12, however, must have 4 (two factors that are 2s) and 3 as factors, and applying these two tests will work as a divisibility test for 12. Thus, divisibility tests that you know can help you develop new tests. We say that 4 and 3 are relatively prime because they have no prime factor in common, or equivalently, their only common factor is 1. Discussion 3 Are They Relatively Prime? Are 12 and 6 relatively prime? Are 15 and 6 relatively prime? Are 25 and 6 relatively prime? Are 7 and 11 relatively prime? Are any two prime numbers relatively prime? General divisibility tests to test for composite factors m. n: If a number p is divisible by m and also by n, and if m and n are relatively prime, then the number p is divisible by the number m. n. This general rule can be used to construct new divisibility tests similar to what we discussed for 6 and for 12. Divisibility tests can help us find the prime factorizations of larger numbers. Consider n = 12,320; what is its (unique) prime factorization?

18 246 Chapter 11 Number Theory We know that 2 divides the number because it ends in 0. We know 5 divides the number because it ends in 0. So we know that 2. 5 = 10 is a factor of 12,320 (and you may have noticed that immediately). Thus we know that 12,320 = We also know that 4 divides the number 1232 because the final two digits form the number 32, which is divisible by 4. A little division then shows 1232 = , so we now have 12,320 = , or But 4 is also a factor of 308 (since 8, from 08, is divisible by 4). So 12,320 = Because the sum of the digits of 77 is 14, which is not divisible by 3, we know 77 and n are not divisible by 3. Nor are 2, 4, 5, or 10 factors of 77. But 77 is 7. 11, so we finally have 12,320 = = , which could be written more compactly using exponents as 12,320 = Activity 12 A New Way of Finding Prime Factorizations Use the method explained here to find the prime factorizations of 1224; of The divisibility tests for the prime numbers 2, 3, 5 are not difficult, and less simple ones for 7 and 11 exist. However, for some primes it is easier to simply divide by the prime and notice whether the quotient is a whole number, than to use complicated and hard-to-remember tests. Suppose the divisibility test for 2 tells you that 2 is not a factor of some number n; could 4 nonetheless be a factor of n? One way to consider this is as follows: If 2 is not a factor of n, 2 cannot appear in the prime factorization of n. But if 2 cannot appear in the prime factorization, 4 could not be a factor of n, because then that 4 could give 2 as a factor (twice) in the prime factorization resulting from having 4 as a factor. (Recall that there can be only one prime factorization of n.) So if 2 is not a factor, then 4 cannot be a factor either. THINK ABOUT... Give a similar argument to convince yourself that if 3 is not a factor of n, then 6 cannot be a factor of n. Give an argument that if p is not a factor of n, then k. p is not a factor of n. Discussion 4 True or False? Discuss whether each of the following is true. Explain your answers, giving counterexamples for false statements. a. If 7 is not a factor of n, then 14 is not a factor of n. b. If 7 is a factor of n, then 14 is a factor of n. c. If 14 is not a factor of n, then 7 is not a factor of n. d. If 20 is not a factor of m, then 60 is not a factor of m.

19 Section 11.3 Divisibility Tests to Determine Whether a Number Is Prime 247 e. If 20 is a factor of m, then 60 is a factor of m. f. If 60 is not a factor of m, then 20 is not a factor of m. g. If a number is a factor of n, then the number is a factor of any multiple of n. h. If a number is a factor of a multiple of n, then the number is a factor of n. i. If a number is not a factor of n, then the number is not a factor of any (nonzero) multiple of n. j. If a number is not a factor of a multiple of n, then the number is not a factor of n. What the above means for testing for primeness is this important fact: You need test only for divisibility by primes when deciding whether or not a number is prime. If 7, say, is not a factor, then 14 or 21 or 28, etc., will not be factors either. You would be wasting your time in testing whether 14 or 21 or 28 were factors, once you found out that 7 was not a factor. In trying to determine whether 187 is a prime then, you would need to find out whether any one of 2, 3, 5, 7, etc., is a factor if one is, you have found a third factor (besides 1 and 187), and so 187 would be composite. But if a prime such as 2 or 3 or 5 or 7 is not a factor, then you do not need to think about their multiples being factors. The following discussion allows us to refine a rule in testing a number for primeness. The issue is, how many primes do you have to test in deciding whether a number like 661 is a prime? If you find that a prime like 2 or 3 or 5 or 7 is a factor of 661, you are done, of course. The number 661 would not be a prime. Discussion 5 Testing for Primes Check to see whether 661 is divisible by 2, by 3, by 5, by 7, by 11. To test whether or not 661 is prime, how many more primes do you think you need to test? Do you think you would need to test for divisibility by 91? Why or why not? You may have found that you needed to test for divisibility by more primes than 2, 3, 5, 7, or 11 to find the prime factorization of 661. But as a matter of fact, you did not have to test for any prime factors greater than 23. This fact comes from the following: When testing whether or not n is prime, you need test only the primes n. This is because if both p > n and q > n, then pq > n. Thus, either p or q must be n if pq = n. Hence, in trying to find out whether 661 is a prime, one would at worst have to try only the primes less than or equal to 661, which is about 26. Primes less than 26 are: 2, 3, 5, 7, 11, 13, 17, 19, and 23. If none of these is a factor, then 661 is a prime. Even if you have a calculator handy, it is good practice to zero in on the square root of a number with educated efforts at trial and error. For example, for 661 think 20 2 is 400 so 20 is too small, and 30 2 is 900 so 30 is too large. Check 25 2 ; 625 is slightly

20 248 Chapter 11 Number Theory less than 661, and 29 2 will be much larger than 661, so checking for primes less than 25 is sufficient. THINK ABOUT... What is the largest prime you need to worry about to find out whether 119 is prime? What about 247? TAKE-AWAY MESSAGE... You now have divisibility tests for 2, 3, 4, 5, 6, 8, 9, 10, 12, and others that have two relatively prime factors such as 15, 18, etc. For primes other than those listed, dividing by the prime can be used to see if the prime is a factor. Using these tests can simplify the work of finding out whether or not a number is prime. You need find only one divisor for n (other than 1 and n) to show that a number is not prime. Moreover, you need not check for any primes larger than the square root of the n to determine whether or not a number is prime. Learning Exercises for Section Practice the divisibility tests for 2, 3, 4, 5, 6, 8. 9, and 10 on these numbers: a b c d Give a six-digit number such that a. 2 and 3 are factors of the number, but 4 and 9 are not. b. 3 and 5 are factors of the number, but 10 is not. c. 8 and 9 are factors of the number. 3. What could a be in n = 4187a432, if 3 is a factor of n? If 9 is a factor of n? Give all the single-digit possibilities for a. 4. Notice that 2 and 4 are factors of 12, but 2. 4, or 8, is not. So a divisibility test for 8 that is NOT safe is to use the 2 test and the 4 test. Find counterexamples for these plausible-looking, but unreliable, divisibility tests. a. 12 is a factor of n if and only if 2 is a factor of n and 6 is a factor of n. b. 18 is a factor of n if and only if 3 is a factor of n and 6 is a factor of n. c. 24 is a factor of n if and only if 4 is a factor of n and 6 is a factor of n. 5. Try these conjectured divisibility tests with 3 or 4 examples each. a. 10 is a factor of n if and only if 2 is a factor of n and 5 is a factor of n. b. 12 is a factor of n if and only if 3 is a factor of n and 4 is a factor of n. c. 18 is a factor of n if and only if 2 is a factor of n and 9 is a factor of n. d. 24 is a factor of n if and only if 3 is a factor of n and 8 is a factor of n. Examine Exercises 4 and 5 to see whether you can predict when such test-twofactors approaches will work, and when they will not. 6. Using Exercise 5, determine whether a. 24 is a factor of

21 Section 11.3 Divisibility Tests to Determine Whether a Number Is Prime 249 b. 24 is a factor of c. 18 is a factor of d. 18 is a factor of e. 45 is a factor of Using Exercise 5, find a 15-digit number that is a multiple of 36; a 15-digit number that is not a multiple of The divisibility tests given here depend on the number being expressed in base ten. The tests are properties of the numeration system rather than of the numbers. Find examples with numbers written in base five to show that, say, the (base-ten) divisibility test for two does not work in base five. You will want to find a number for which two is (or is not) a factor but whose base five representation does (or does not) satisfy the divisibility test for two that you know for base ten. 9. Explain why finding only one factor of n besides 1 and n is enough to show that n is composite. 10. Suppose that n = Give the prime factorization of n 2 and n 3. (Hint: Do not work too hard.) 11. Determine whether each of these is a prime. a. 667 b. 289 c d e f The numbers 2 and 3 are consecutive whole numbers, each of which is a prime. Is there another pair of consecutive whole numbers, each of which is a prime? If there is, find such a pair; if not, explain why. 13. Test each of these numbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, and 18. a. 540 b. 150 c. 145 d. 369 e Which of these numbers are prime? For those not prime, give the prime factorization. a b c. 127 d. 121 e f g Here is an interesting conjecture that mathematicians are uncertain about even though it has been studied for more than 100 years: Every even number greater than 2 can be written as the sum of two primes (Goldbach s conjecture). Test the conjecture for the even numbers through Devise a way of checking to see whether or not a number is divisible by 24. Test your method on 36; on Are every two different primes relatively prime? Explain. 18. Which pairs of numbers are relatively prime? a. 2, 5 b. 2, 4 c. 2, 6 d. 2, 7 e. 2, 8 f. 2, 9 g. 8, 9 h. 8, 12 i. 3, 8 j. 40, 42 k. 121, 22 l. 39, Give an example of two 3-digit nonprime numbers that are relatively prime. 20. If possible, give a composite number that is relatively prime to 22.

22 250 Chapter 11 Number Theory 21. a. Find and check your answer by casting out nines. b. Compute and check your answer by casting out nines. 22. A result in number theory states that the product of any n consecutive positive integers is divisible by the product of the first n positive integers. For example, = The theorem asserts that 1680 is divisible by , or 24. Verify that 1680 is divisible by 24, using divisibility rules. Compute the product of another 4 consecutive positive integers and check to see if the product is divisible by 24. Demonstrate this theorem for some example you choose for n = A result in number theory states that if p is a prime number and n is a positive integer, then n p n is divisible by p. Demonstrate this for 3 cases where you choose n and p. 24. a. I am a 3-digit number. I am not a multiple of 2. 7 is not one of my factors, but 5 is. I am less than 125. Who am I? b. Make up a Who am I? involving number theory vocabulary and ideas Greatest Common Factor, Least Common Multiple As you would suspect about an old area of mathematics like number theory, there are entire books on the subject, exploring many different and advanced areas. So the work with number theory in the elementary school curriculum touches on only a small part of number theory. Elementary school number theory usually comes up right before work with fractions. Simplifying fractions and finding common denominators for adding and subtracting fractions use number theory ideas. The same ideas carry over to algebraic fractions, so even though fraction calculators might be available, the reasoning behind the work with regular fractions will continue to be important. Recall from Chapter 6 that a basic result about fractions is that 1 a b = a b. The number 1 can be written as n n, where n is any nonzero number. So, n. n a b = na nb. Starting with a b, this allows one to generate any number of fractions equal to a b simply by making different choices for n. Read backwards, however, na nb = a b, and the result shows that one can write a simpler fraction for a given fraction by finding a common factor of the numerator and denominator, and canceling it out, as you might have said in earlier work.

23 Section 11.4 Greatest Common Factor, Least Common Multiple 251 For example, because 3 is a common factor of 84 and 162, = , so = 28 54, an equivalent but simpler fraction. But you should notice that can be simplified further, because 2 is a common factor of 28 and 54: = Because 14 and 27 have only 1 as a common factor (they are relatively prime ), is the simplest form for If you can find the greatest common factor (GCF, also called the greatest common divisor), of the numerator and denominator (which is 6), then you can get the simplest fraction in one step: = = For reasonably small pairs of numbers, you can often see the greatest common factor by inspection. A systematic way would involve listing all the factors of each number and then picking out the greatest common factor. Set of factors of 84 = {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84} Set of factors of 162 = {1, 2, 3, 6, 9, 18, 27, 54, 81, 162} A Venn diagram is one way of illustrating the common factors, 1, 2, 3, and 6. 4, 7, 12, 14, 21, 28, 42, 84 1, 2, 3, 6 9, 18, 27, 54, 81, 162 Factors of 84 Factors of 162 The region they share provides common factors; 1, 2, 3, and 6, and so 6 is the GCF of 84 and 162. Sometimes a more efficient method would be to examine the factors of the smaller number to see which are also factors of the larger number. Usually you start with the larger factors because you are looking for the greatest common factor. For 84, you would try 84, 42, etc., working either from the complete list or hoping not to overlook a larger common factor than any you find. Activity 13 GCFs Find the prime factorizations of 84 and 162. What do they share in common? Knowing how to find the least common multiple of fraction denominators can make the adding or subtracting of fractions easier. As you may recall, to find the sum of fractions like 9 16 and 7 12, the usual algorithm or method calls for replacing the given fractions with fractions that have the same denominators but are equal to the original

24 252 Chapter 11 Number Theory fractions ( equivalent fractions with a common denominator might be the language used). One can always multiply the numerator and denominator of each fraction by the other denominator ( and 16 12, or 192 and 192 ) but usually those new fractions do not lead to the simplest arithmetic. Finding the least common denominator is the usual approach, just to keep the numbers smaller. This least common denominator is just the least common positive multiple of the denominators of the fractions involved. As with the greatest common factor, finding the least common multiple can be approached in several ways. One way, perhaps best when the idea is new, is to list all the common multiples of each number until a common one (not 0) is found. (0 is literally the least common multiple of every two whole numbers, but it is not useful in situations where least common multiples arise, as with fractions.) Set of multiples of 16 = {0, 16, 32, 48, 64, 80, 96, 112, 128,...} Set of multiples of 12 = {0, 12, 24, 36, 48, 60, 72, 84, 96, 108,...} A Venn diagram can also be used to illustrate the multiples: 16, 32, 64, 80, , 48, , 24, 36, 60, 72, 84, Multiples of 16 Multiples of 12 Because 48 is the least common nonzero multiple of 16 and 12, 48 would be the smallest number that could serve as a common denominator in adding, for example, = , or = The sum is not always in simplest form, but if you are calculating by hand, the fractions with the least common multiple (LCM) as denominator offer the simpler arithmetic with the smaller numbers. Activity 14 Using LCMs and GCFs 1. What is the GCF of 68 and 102? 2. Simplify What is the LCM of 68 and 102? 4. What is ?

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