1. Copyright © Cengage Learning. All rights reserved. CHAPTER 4 Number Theory

2. Copyright © Cengage Learning. All rights reserved. SECTION 4.1 Divisibility and Related Concepts

3. 3 What Do You Think? What do the terms odd and even mean? How can you tell whether one number is divisible by another without dividing? What connections do you see between divisibility and decomposition?

4. 4 Divisibility and Related Concepts In this section, we will investigate the basic concepts that will help you to understand the various structures of natural numbers. There are two different ways in which numbers can be decomposed: additive and multiplicative. For example, 12 can be decomposed into and it can be decomposed into 10 + 2, and it can be decomposed into 4  3. Depending on circumstances, either decomposition might be relevant.

5. 5 Divisibility and Related Concepts In this chapter, we will explore multiplicative decompositions of natural numbers. As you will discover, being able to decompose numbers and knowing which decomposition is relevant are themes that recur throughout elementary mathematics.

6. 6 Investigation A – Interesting Dates We will begin our investigations with a playful question. Many mathematics teachers and mathematicians would smile when writing down December 8, 2096. If we were to write this date in shorthand, we would write 12/8/96. A. Describe the relationship among these three numbers in words. B. Determine how many instances of this pattern will occur in the years 2000–2019.

7. 7 Investigation A – Discussion A. One way of describing the relationship among the three numbers is to say that the third number is the product of the first two: 96 is the product of 12 and 8. There are several mathematical concepts and terms that come out of this relationship: factor, multiple, divisible, divisor, and divides. These terms are all related. Therefore, we will define one term formally and state how the other terms are related to it.

8. 8 Investigation A – Discussion If a and b are two whole numbers (b  0) and there is a third natural number c such that a  c = b, then we say that a divides b, and we write a | b. When one number does not divide another number, we write a b. If a | b, then a is a factor of b. If a | b, then b is a multiple of a. If a | b, then b is divisible by a. If a | b, then a is a divisor of b. cont’d

9. 9 Investigation A – Discussion B. Let us now explore the second question. There are no such dates in 2000. In 2001, there is only one such date: 1/1/01. Note: Table 4.1 is not the only possible table. cont’d Table 4.1

10. 10 Investigation A – Discussion In one sense, we are done—we have found all 53 possible dates. However, there is still a lot of interesting mathematics to be gleaned from looking for and examining patterns and relationships. Take a few minutes to examine the table 4.1. What patterns and relationships do you observe in the table? One pattern is that many dates have a twin, for example, 1/2/02 and 2/1/02, and 2/3/06 and 3/2/06. Mathematically, we can talk about symmetry in our set. cont’d

11. 11 Investigation A – Discussion We can also see that some dates do not have twins. For example, 1/7/07 has a twin, but 1/13/13 doesn’t. If we stop for a moment, we realize that 13/1/13 is not a valid date. Some valid dates have no twins: 2/2/04, 3/3/09, 4/4/16. In this case, the numbers 4, 9, and 16 are square numbers. cont’d

12. 12 Odd and Even Numbers

13. 13 Odd and Even Numbers One of the first ways in which children encounter number theory is with odd numbers and even numbers. Let me introduce this discussion with a story. One morning my five-year-old son, Josh, asked me at the breakfast table if 44 was an odd or an even number; at that time, my 45th birthday was a few days away. I asked him what he thought, and he said he thought it was even. I asked him what he thought an even number was.

14. 14 Odd and Even Numbers Josh’s response was that even numbers were fair. I asked him what he meant by fair. His response was, “Well, it’s fair if two teams can get the same amount.” Take a few moments to write your own definition of odd and even. Below are several different ways in which even numbers have been defined. Note that the notion of even is closely tied to the notion of decomposition.

15. 15 Odd and Even Numbers That is, if a number can be decomposed in the following way, then it is even. A whole number is an even number if: It has 2 as a factor. It is divisible by 2. Its ones digit is 2, 4, 6, 8, or 0. (This characteristic does not hold for all bases; do you see why?) It can be represented as the sum of two equal whole numbers. There exists another whole number a such that x = 2a.

16. 16 Investigation B – Patterns in Odd and Even Numbers In this investigation, we will examine and discover some of the many patterns that emerge when odd and even numbers are combined. There are several reasons for investigating odd and even numbers. One is that many people find it fun; I have seen the excitement on young children’s faces as they discover these patterns. Another reason is that in order to explain these patterns, one has to look more closely at the concepts.

17. 17 Investigation B – Patterns in Odd and Even Numbers Begin the investigation by thinking about the following questions A. What do you notice about adding two odd numbers? Two even numbers? An odd number and an even number? B. Can you explain the pattern you see in the sum of two odd numbers? cont’d

18. 18 Investigation B – Discussion A. The patterns for adding odd and even numbers can most succinctly be represented as odd + odd = even even + even = even odd + even = odd B. Now that we have seen what the pattern is, how might we explain why it holds for all odd and even numbers? What tools do you have that might enable you to show why this is true?

19. 19 Investigation B – Discussion Strategy 1: Make a Drawing Let us begin by examining the question from a geometric perspective. Figure 4.3 represents any odd number. Why is this figure a valid representation of any odd number? cont’d Figure 4.3

20. 20 Investigation B – Discussion Think of two Cuisenaire rods, one of which is one unit longer than the other. Any odd number can be represented in this manner. Can you connect any of your definitions to this diagram? How? If we think of even as meaning able to be separated into two groups that have the same amount, then odd must mean that we can’t do that; in other words, one of the rods will be one unit longer. What if we combine two different odd numbers? Draw that diagram. What do you see? cont’d

21. 21 Investigation B – Discussion When we build a concrete model to show the combination of two odd numbers, the first odd number has one unit left over and the second odd number has one unit left over, as in Figure 4.4. When we combine the two numbers, the two units become a pair. cont’d Figure 4.4

22. 22 Investigation B – Discussion There are two ways in which this model connects with our idea of even. Some students might say that now there are no units left over, so we have an even number. Other students might use language from our work with base ten blocks and say that when we combine the two numbers, we have two sticks of equal length; thus, it is an even number. cont’d

23. 23 Investigation B – Discussion Strategy 2: Use Algebra Let us now examine an algebraic explanation for the fact that the sum of two odd numbers must be an even number. Because one of the characteristics of all even numbers is that they are divisible by 2, we can say that 2n represents any even number as long as n represents a natural number. Because one of the characteristics of all odd numbers is that they are not divisible by 2, we can say that 2n + 1 represents any odd number. cont’d

24. 24 Investigation B – Discussion However, this form means that the number 2(n + m + 1) has to be an even number. cont’d

25. 25 Investigation C – Understanding Divisibility Relationships There are a host of theorems that concern divisibility. We will examine two here. A. We know that 6 | 42 and 6 | 72. Does it necessarily follow that 6 divides their sum—that is, does 6 divide (42 + 72)? Discussion: In this particular case, it is true; 6 does divide 42 + 72, because 6  19 = 114.

26. 26 Investigation C – Understanding Divisibility Relationships B. In fact, pick any three natural numbers a, b, and c for which a | b and a | c. Then a | (b + c). Can you prove this? Let me restate this general question in basic English and in mathematical notation: English: If a number divides two numbers, will it necessarily divide their sum? Notation:If a | b and a | c, is it always true that a | (b + c)? How might we prove this? cont’d

27. 27 Investigation C – Discussion One proof uses the definition of divisibility and looks like this: If a | b, then there is a natural number x such that ax = b. Similarly, if a | c, then there is a natural number y such that ay = c. The proof consists of transforming b + c to show that it must be divisible by a: b + c = ax + ay = a(x + y) Here we are substituting ax for b and ay for c. This is true because of the distributive property.

28. 28 Investigation C – Discussion We have proved the general case, because if b + c can be expressed as the product of a and some natural number, then by definition (b + c) is divisible by a. Therefore, we have proved that if a | b and a | c, then a | (b + c). Two theorems similar to this one, are as follows: If a | b and a | c, then a | (b – c) If a | b and a | c and a | d, then a | (b + c + d) cont’d

29. 29 Investigation D – Determining the Truth of an Inverse Statement Suppose we turn things around and consider the inverse statement: What if a number a divides neither of two larger numbers? Can we say that it will never divide the sum of those two numbers? This question can be stated in basic English and in notation as follows: English: If a does not divide b and a does not divide c, then will a not divide their sum? Notation: Is it always that if a b and a c, then a (b + c)?

30. 30 Investigation D – Discussion Although this hypothesis seems reasonable, we can show that it is invalid by using a counterexample. For example, consider a = 3, b = 7, and c = 2. Although 3 7 and 3 2, 3 | (7 + 2). Thus it is not true that if a b and a c, then a (b + c). Note: A mathematical statement is considered to be true only if it is true 100 percent of the time. If there is even one exception (a counterexample), then the statement is considered to be mathematically false.

31. 31 Divisibility Rules

32. 32 Divisibility Rules Before the widespread use of calculators, knowing divisibility rules was quite useful. For example, simplifying is much easier for a student who immediately sees that both numbers are divisible by 6. Several of the divisibility rules are quite simple. Divisibility by 2 When will a number be divisible by 2? Another way of asking this question is: If you think of all numbers that are divisible by 2, what do they have in common?

33. 33 Divisibility Rules What they have in common is that they are all even numbers. Thus the divisibility rule for 2 is A whole number is divisible by 2 iff it is an even number. An equivalent statement is A number is divisible by 2 iff its ones digit is a 0, 2, 4, 6, or 8. Note that this statement holds for base ten.

34. 34 Divisibility Rules Divisibility by 5 This rule is simple in base ten: A whole number is divisible by 5 iff its ones digit is a 0 or a 5. Think back to base ten manipulatives. When we count by 5s, we are counting by half longs. Divisibility by 10 A whole number is divisible by 10 iff the ones digit is 0.

35. 35 Divisibility Rules These three divisibility rules are generally the easiest to remember. Do you think it is a coincidence that 2 and 5 both divide 10? Divisibility by 3 Make up several numbers and determine whether they are divisible by 3. Look at the numbers that are divisible by 3.

36. 36 Investigation E – Understanding Why the Divisibility Rule for 3 Works Formally, we say that A whole number is divisible by 3 iff the sum of the digits of the number is divisible by 3. For example, 3 | 567 because 3 | (5 + 6 + 7). Let us use the divisibility relationships that we explored earlier to explain why the divisibility rule for 3 works. Pick a few numbers that are divisible by 3 and see whether you can tell what the sum of the digits has to do with divisibility by 3. For example, 243 is divisible by 3.

37. 37 Investigation E – Discussion Begin with manipulatives We will first work through a concrete representation and then move to the more general case. If we represent 243 with base ten manipulatives, we get Figure 4.5. Figure 4.5

38. 38 Investigation E – Discussion In one sense, we can look at this representation as three distinct groups: flats, longs, and units. The key to understanding why this rule works lies in the fact that if a | b and a | c, then a | (b + c). We will use an unproved extension of that theorem: If a | b, a | c, and a | d, then a | (b + c + d) for all natural numbers a, b, c, and d. cont’d

39. 39 Investigation E – Discussion If we can break down our number into various components so that each component is divisible by 3, then the number will be divisible by 3. What can we do to the flats and the longs to make them divisible by 3? If we cut one unit from each flat and each long, we still have the same amount—that is, 243 (Figure 4.6). But now the 99 blocks and the 9 blocks are divisible by 3. cont’d Figure 4.6

40. 40 Investigation E – Discussion We simply move the “extra” units created in cutting the flats and longs and put them with the original 3 units (Figure 4.7). Now we have 9 ones, and 9 is divisible by 3. cont’d Figure 4.7

41. 41 Investigation E – Discussion Now use expanded notation We can use expanded notation now to show what we have done and to understand better why the divisibility rule for 3 works. cont’d

42. 42 Investigation E – Discussion Finally, use the theorem We have now decomposed 243 into three addends: 2(99), 4(9), and (2 + 4 + 3). We now use two theorems from number theory: 1. if a | b, then a | bc. 2. if a | b, a | c, and a | d, then a | (b + c + d). Because 3 | 99, it also divides 2(99); because 3 | 9, it also divides 4(9); and we can see that 3 | (2 + 4 + 3). cont’d

43. 43 Investigation E – Discussion Using the second theorem, 3 divides 2(99) + 4(9) + (2 + 4 + 3), which equals 243. Generalizing the justification This does not constitute a proof of the divisibility rule for 3 but is, rather, an examination into the structure of the rule. If we had represented 243 as abc, we would have a proof of the divisibility rule for 3 for any three-digit number. cont’d

44. 44 Investigation E – Discussion That would go like this: abc = 100a + 10b + c = 99a + a + 9b + b + c = 99a + 9b + (a + b + c) cont’d

45. 45 Investigation E – Discussion Divisibility by 9 As you discovered when you worked with the multiplication table, the 3 family and the 9 family are related. So too are their divisibility rules. Before reading the rule below, see whether you can guess what it is using either inductive reasoning (making up actual numbers) or deductive reasoning. A whole number is divisible by 9 iff the sum of its digits is divisible by 9. cont’d

46. 46 Investigation F – Divisibility by 4 and 8 Just as the divisibility rules for 3 and 9 are similar, so too are the divisibility rules for 2, 4, and 8. A. As before, choose several numbers and determine which are divisible by 4. Then find patterns that will lead you to be able to predict whether a number is divisible by 4 without dividing. B. Choose several numbers and determine which are divisible by 8. Find patterns to help you predict whether a number is divisible by 8.

47. 47 Investigation F – Discussion A. The divisibility rule for 4 is not terribly obvious. Let me offer a hint before presenting the rule. Consider the number 532. The test for divisibility by 4 centers on whether 4 | 32. Similarly, with the number 123,456, the test for divisibility by 4 centers on whether 4 | 56. There are several valid ways to express the rule. One looks like this: A whole number is divisible by 4 iff the number represented by the last two digits is divisible by 4. By the last two digits, I mean the ones digit and the tens digit.

48. 48 Investigation F – Discussion The key to understanding how this rule works is to realize that no matter how many hundreds we have, that part of the number will always be divisible by 4. This is true because 4 | 100, and therefore any multiple of 100 will also be divisible by 4. Proof for three-digit numbers: cont’d

49. 49 Investigation F – Discussion B. To help you understand the rule rather than just memorizing it, consider the number 2328 and whether it is divisible by 8. This is another case in which expanded notation is helpful: 2328 = 2000 + 300 + 20 + 8 Will any of the parts be divisible by 8? cont’d

50. 50 Investigation F – Discussion Because 8 | 1000, any multiple of 1000 will be divisible by 8, so 8 | 2000. However, 8 does not divide 100, so we need to look at the last three digits to determine divisibility by 8. That is, if 8 | 238, then the original number, 2328, will be divisible by 8. The divisibility rule for 8 uses language similar to the rule for 4: A whole number is divisible by 8 iff the number represented by the last three digits is divisible by 8. cont’d

51. 51 Divisibility Rules Divisibility by 6 The divisibility rule for 6 has a different flavor from the other rules. How might you determine whether a large number is divisible by 6 without actually dividing? A whole number n is divisible by 6 iff n is divisible by 2 and by 3. This can also be stated as 6 | n iff 2 | n and 3 | n Let us now investigate how we might apply this divisibility rule to a larger number.

52. 52 Investigation G – Creating a Divisibility Rule for 12 What about a divisibility rule for 12? Can we use what we know, or do we just have to divide (longhand or with a calculator)? Discussion: When I have asked my classes this, I generally get different hypotheses. Some students say that if a number is divisible by 2 and by 6, then it must be divisible by 12. Other students say that if a number is divisible by 3 and by 4, then it must be divisible by 12. Is this one of those cases in which both answers are correct, or is only one correct?

53. 53 Investigation G – Discussion In this case, the first hypothesis is not always true. Can you see why? For example, 2 | 18 and 6 | 18 but 12 18. If you thought this first hypothesis was correct, can you now see why it isn’t? If you knew that it wasn’t correct, can you explain why? This hypothesis doesn’t work because “divisible by 2” is a redundant condition. If a number is divisible by 6, then it must be divisible by 2. Thus, saying that a number is divisible by 2 and divisible by 6 is mathematically equivalent to saying simply that it is divisible by 6. cont’d

54. 54 Divisibility Rules Revisiting Venn diagrams The preceding sentence makes great sense to some students and “sort of” sense to others. If you are in the latter set, the old adage that a picture is worth a thousand words may be relevant. Let us examine some Venn diagrams and see how they can contribute to our understanding of divisibility rules.

55. 55 Divisibility Rules To say that if a number is divisible by 6, it must be divisible by 2 means that the set of numbers divisible by 6 is a subset of the set of numbers divisible by 2, as shown on the left in Figure 4.8. Figure 4.8

56. 56 Divisibility Rules When we look at divisibility by 3 and by 4, however, we see that they are not disjoint sets (like the sets of even and odd numbers), but rather overlapping sets. The diagram on the right in Figure 4.8 illustrates the relationship between the numbers divisible by 3 and the numbers divisible by 4. The intersection of these two sets is the set of numbers divisible by 12. Thus we can say that a number is divisible by 12 iff it is divisible by 3 and by 4.