Learning Module 01 Integer Arithmetic and Divisibility 1 1 Objectives. Review the properties of arithmetic operations. Understand the concept of divisibility. Understand the formal definitions of quotient and remainder. Review some factorization formulas. 2 Integer Arithmetic. Number theory is the area of mathematics that studies the properties of the integer numbers: Z t..., 3, 2, 1,0,1,2,3,...u Given two integer numbers a and b, we can always compute their sum a b, difference a b and product ab. On the other hand, the division of a by b is not always defined. Addition and multiplication have the usual elementary properties that we learn in school. Using these properties it is possible to prove many algebraic identities. For example, we can make the following sequence of deductions: pa bq 2 pa bqpa bq pa bqa pa bqb aa ba ab bb Definition of squared. Distributive property. Distributive property, again. a 2 2ab b 2 Gathering common terms. We will use these well known formulas freely in the text (some of them are reviewed in section 3. Readers of this text are assumed to be capable of providing these kinds of proof whenever it is required. This text attempts to be as formal as adequate for an introductory course in number theory, but is not intended to be course on the foundations of number systems. It is, on the other hand, useful to make connections with more advanced concepts, mainly those of abstract algebra. 3 Divisibility, quotient and remainder. Divisibility is a central notion in number theory. Informally, an integer b 0 divides a if a{b is an integer, but it is more convenient to write definition a little differently. Definition 3.1. Let a and b are arbitrary integers. Then we say say that b is divides a if there is an integer q such that a qb. In this case, we also say that b is a divisor or factor of a, and that a is a multiple of b. We use the following notation: ba means that b divides a; b a means that b does not divide a. For example, 728, since 28 4 7. Also, 3111, because 111 p 37q p 3q. On the other hand, 4 38 and 35 1001. Note: it is important to realize that, unless otherwise stated, integer numbers in definitions and theorems are allowed to be positive, negative or zero.
Learning Module 01 Integer Arithmetic and Divisibility 2 The number 0 is special for divisibility: every number b divides 0, since 0 0 b, but the only number that divides 0 is 0 itself (00). In one of the examples above, we mentioned that 35 1001. So, it is not possible to write 1001 as an exact multiple of 35. However, in grade school we learn that when 1001 is divided by 35, we get a quotient of 28 and a remainder of 21. This means that: 1001 28 35 21. This equation, however, by itself does not characterize quotient and remainder. It is also true that 1001 26 35 91, and we don t say that 26 is the quotient of the division of 1001 by 35. The problem is that the remainder is not as small as possible in the latest equation we can still subtract 35 from 91 and obtain a non-negative result. This is the idea we use to define quotient and remainder. We have to do things a little differently, however, because we want to write a definition that works for negative numbers as well. Our definitions are based on the following result. Theorem 3.2. Let a and b be integers, where b 0. Then there are integers q and r such that: and Furthermore, the q and r satisfying (3.1) and (3.2) are unique. a qb r (3.1) 0 r b and rb 0. (3.2) Notice that the condition rb 0 means that divisor and remainder always have the same sign. We use the following notation: a div b is the quotient of a divided by b; a mod b is the remainder of a divided by b. Example 3.3. What are the quotient and remainder of 145 divided by 33? We have: 145 p 5q p 33q 20. Also, the divisor, 33, and 20 have the same sign. Finally, 20 33. We conclude that 145 divided by 33 gives a quotient of 5 and a remainder of 20. Equivalently: 145 div p 33q 5, 145 mod p 33q 20. This may seem confusing if you have never done integer division with negative numbers. There is, however, a very simple way of computing quotient and remainder. We first define to very important functions. Definition 3.4. Let x be a real number. Then: 1. The floor of x, denoted by txu, is the largest integer that is not strictly above a. 2. The ceiling of x, denoted by rxs, is the smallest integer that is not strictly below a. For example t3.2u 3, t 3.9u 4, r4.3s 4, r 0.8s 0. Then we have:
Learning Module 01 Integer Arithmetic and Divisibility 3 Theorem 3.5. Let a and b 0 be integers. Then the quotient and remainder of a divided by b are given by:. q ta{bu (3.3) r a qb (3.4) Example 3.6. Let s compute quotient and remainder for a 145 and b 33, using formulas (3.3) and (3.4). We have: Z 145 div p 33q 145 ^ t 4.393939...u 5, 33 145 mod p 33q 145 p 5q p 33q 20. 4 Factorization Formulas In mathematics, it is often necessary to write formulas in shorter, more understandable, forms. For example, the expression: 4a 2 12a 9 can be written more compactly as: p2a 3q 2. These are some of the more important factorization patterns: Let s go over a few examples: a 2 2ab b 2 pa bq 2 (4.1) a 2 2ab b 2 pa bq 2 (4.2) a 2 b 2 pa bqpa bq (4.3) a 3 b 3 pa bqpa 2 ab b 2 q (4.4) a 3 b 3 pa bqpa 2 ab b 2 q (4.5) Example 4.1. Factor 9a 2 24a 16. We start by noticing that 9a 2 p3aq 2 and 16 4 2, so our expression seems to fit the pattern of formula (4.1). We then compute: 2p3aqp4q 24a, so that the middle term also agrees with the pattern. We conclude that: 9a 2 24a 16 p3a 4q 2. Example 4.2. Factor 8x 3 343y 3. We have 8x 3 p2xq 3 and 343y 3 7y 3, so our expression seems to fit the pattern of formula (4.5). Next, we compute: p2 xqp7 yq 14 xy. Thus: 8x 3 343y 3 p2x 7yqp4x 2 14xy 49y 2 q. Example 4.3. Factor the expression a 4 81 as completely as possible. This expression can be viewed as a difference of squares (formula (4.3)): a 4 81 pa 2 9qpa 2 9q. Now, a 2 9 is itself a difference of squares: a 2 9 pa 3qpa 3q. Finally, we have: a 4 81 pa 2 9qpa 3qpa 3q
Learning Module 01 Integer Arithmetic and Divisibility 4 We finish this section with a generalization of formulas (4.3), (4.4) and (4.5). Theorem 4.4. Let a and b be integers, and n a positive integer. Then: a n b n pa bqpa n 1 a n 2 b a n 3 b 2 a 2 b n 3 ab n 2 b n 1 q (4.6) Also, if n is odd: a n b n pa bqpa n 1 a n 2 b a n 3 b 2 a 2 b n 3 ab n 2 b n 1 q (4.7) Example 4.5. Let s factor 32x 5 1. We have 32x 5 p2xq 5, and 1 1 5. Since n 5 is odd, we can use formula (4.7): 32x 5 1 p2x 1qpp2xq 4 p2xq 3 p2xq 2 2x 1q p2x 1qp16x 4 8x 3 4x 2 2x 1q Example 4.6. Sometimes, different strategies provide different factorizations. In these cases, it is always preferable to use the most complete factorization. Consider the expression a 6 729. Since 729 3 6, formula (4.6) gives: a 6 729 pa 3qpa 5 3a 4 9a 3 27a 2 81a 243q. However, it is also possible to consider the expression as a difference of squares: a 6 729 pa 3 q 2 p3 3 q 2 pa 3 3 3 qpa 3 3 3 q. Then we can use the formulas for the sum and difference of cubes to get (after rearranging terms): a 6 729 pa 3qpa 3qpa 2 3a 9qpa 2 3a 9q. Example 4.7. The formulas for sum and difference of powers are useful for finding (nontrivial) divisors of some large integers. Let s find a factor of u 535537 531441 4096. Whenever facing this kind of problem, readers are advised to keep their eye out for tricks. We notice that 531441 3 12 and 4096 2 12, so that u 3 12 2 12. We can t use formula (4.7) directly, because the exponent is even. However, we can write u p3 4 q 3 p2 4 q 3, and by formula (4.7), one factor of u is given by 3 4 2 4 81 16 97. This example is admittedly artificial, but we will see that, when it comes to factoring large numbers, clever manipulations like these sometimes save the day! 5 Problems 1. Prove the following identities: (a) pa bq 3 a 3 3a 2 b 3ab 2 b 3. (b) pa bqp2a bq 2a 2 ab b 2 (c) pa b cqpa b cq a 2 b 2 2bc c 2 2. Let a, b, c be integers. Prove the following statements concerning the divisibility relation: (a) If ab and bc then ac. (b) If ab, then ap bq, p aqb, p aqp bq and a b. (c) If ab and ac, then ab c. (d) If ab and c is an arbitrary integer, then abc.
Learning Module 01 Integer Arithmetic and Divisibility 5 Solution to part (a): These are proved by what is called chasing definitions : we have juggle back and forth with the definition of divisibility. For part (a), the hypotheses ab and bc imply that there are integers q 1 and q 2 such that b q 1 a and c q 2 b. Then, c q 2 pq 1 aq pq 1 q 2 qa, and if we let q q 1 q 2 we get c qa. This means that ac, which is the conclusion we wanted to prove. 3. Draw a plot of the functions x Ñ txu and x Ñ rxs for x in the interval r 4,4s. 4. For each pair of integers a, b given below, use Theorem 3.5 to compute a div b and a mod b. (a) a 430 and b 32. (b) a 75 and b 13. (c) a 978 and b 46. (d) a 710 and b 7. 5. Draw a plot of the function x Ñ x txu for x in the interval r 4,4s. This is called the fractional part function 6. Prove the following characterizations of the floor and ceiling of an arbitrary real number x: (a) txu=a if and only if a is an integer such that a x a 1. (b) rxs=a if and only if b is an integer such that a 1 x a. Hints for part (a): It is easier to prove an if and only if statement by separating it in two parts. First assume that txu a. Then clearly a is an integer, and you have to show that a x a 1. One way to do this is to assume that these inequalities are not true, and show that a cannot be equal to txu. Then, you have to prove that, if a is an integer and a x a 1, then txu a, that is, that a is the largest integer that does not exceed x. 7. Determine if each of the following statements is true or false, and justify your answer. To show that a statement is true, you have to provide a proof. To show that it is false, it is enough to provide an example where the statement is not satisfied (this is called a counterexample). (a) For every real number x, t2xu 2 txu. (b) For every real numbers x and y, tx yu = txu tyu. (c) If x is a real number and a is an integer, then tx au txu a. (d) For every real numbers x and y, tx yu txu tyu. 8. Let x be a real number. (a) Denote by roundpxq denote the integer that is closest to x. Show that: Z roundpxq x (b) The formula above implies a convention for roundpxq if x is halfway between two integers. What is this convention? (c) Make a plot of the function x Ñ roundpxq. 9. In this problem, you are asked to provide a proof of Theorem 3.2. We assume that a and b 0 are two integers. ^ 1. 2
Learning Module 01 Integer Arithmetic and Divisibility 6 (a) Let q and r be defined by formulas (3.3) and (3.4). Show that these values of q and r satisfy the conditions (3.1) and (3.2). Hint: Problem 6 will be useful. (b) Now assume that (3.1) and (3.2) are valid. Show that q and r must satisfy formulas (3.3) and (3.4). Hint: Same as above. 10. Prove the factorization formulas (4.1) (4.5). 11. Find the most complete factorization, with integer coefficients, of each of the expressions below. (a) 4a 4 4a 2 b b 2. (b) b 5 16b. (c) 64a 6 b 6 1. (d) 81b 4 1. (e) 2ab 10a 2b 10. 12. Prove formulas (4.6) and (4.7). 13. Find as many divisors as you can of each the integers below, without using a computer. You can use a calculator only to compute powers of integers. Also, it is not necessary to express the answers in decimal form, it is OK to use powers. (a) 10 2009 1. (b) 2401 256 (c) 13122 1250