BASIC CONCEPTS ON NUMBER THEORY

Transcription

1 BASIC CONCEPTS ON NUMBER THEORY This is a brief introduction to number theory. The concepts to cover will enable us to answer questions like: How many integers between 50 and 150 are divisible by 4? How many ending zeroes does 75! have? Which integers have 3 as remainder when divided by 10? What is the least common multiple between 140 and 250? MULTIPLES Let a and b be integers. If a = bk, for some integer k (unique!), we say that a is a multiple of b or b divides a or b is a divisor of a or b is a factor of a. This is denoted it as b a (Do not confuse this notation with b/a which means that b is divided by a). For example, 8 is a multiple of 4 or 4 is a divisor of 8 or 4 divides 8, because 8 = 4 x 2. Hence, we can use the notation 4 8. However, 8 is not a multiple of 5, since 8 5k for any integer k. We can form a set with the multiples of a number. For instance, the set of multiples of 4 is the set {x x = 4k, k Z} = {x x = 4k, k= 0, ± 1, ± 2, ± 3, ± 4,...}= {0, 4, -4, 8, -8, 12, -12, 16, -16,...} That is, as k takes values over all the integer numbers the multiples of 4 are obtained. Observe that zero is a multiple of any number (why?). However, zero does not divide any number (why?). On the other hand, 1 is a divisor of any number (why?). But 1 is not a multiple of any number. REMARK: The set of multiples of 2 is called the set of even numbers. Even Numbers ={,-4,-2, 0, 2, 4, } ={x x=2k, k Z } The set of odd numbers is obtained by adding (or subtracting) 1 to each even number. Odd Numbers = {, -5, -3, -1, 1, 3, 5, } = { x x = 2k+1, k Z } ={ x x = 2k-1, k Z } Discrete Math. Number Theory. Revised 9/98 Page1

2 PRIME NUMBER A positive number greater than 1 is a prime number if it has exactly two positive divisors, 1 and the number itself. A number greater than 1 that is not a prime number is called a composite number. 2, 3, 5 are prime numbers, but 4, 6, 12, 100 are composite numbers. FUNDAMENTAL THEOREM OF ARITHMETIC Every integer greater than 1 is a prime number or it can be written as a product of powers of prime factors. This decomposition is unique except for the order of the factors. EXERCISE 1 10=2*5 8= = is a prime number 1. Choose and even number of even number and add them up. Is the answer even or odd? 2. Do you obtain the same result if you add an odd number of even number? 3. Choose and even number of odd numbers and add them up. Is the answer even or odd? 4. Do you obtain the same answer if you add an odd number of odd number? 5. Generalize your findings from the previous problems (1-4). Show that your results hold in general. 6. Consider the following statements. When the statement is true, show it always holds. Otherwise, give an example to show it is not true. a) A multiple of 6, is a multiple of 3. b) If a number is a multiple of 2 and 5, it is a multiple of 10. c) Any multiple of 4 and 6 is a multiple of 24. d) Any multiple of 12 is a multiple of 6 and Find the prime factorization of Find the prime factorization of 10! (Recall that 10!=10*9*8*...*2*1) 9. How many ending zeroes does 25! have? For example, 750 has one ending zero, has three ending zeroes. LEAST COMMON MULTIPLE Consider the set of positive integer multiples of 4 and the set of positive integer multiples of 6. Call these sets E and F respectively. They are: E = {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48,...} F = { 0, 6, 12, 18, 24, 30, 36, 42, 48,...} The set of common positive integral multiples of 4 and 6 is the intersection of E and F. E F = {12, 24, 36,...} Notice that this set is formed by the positive integral multiples of 12. The least element in this collection is the least common multiple between 4 and 6, and any other common multiple of them is a multiple of 12. We generalize this concept next. Discrete Math. Number Theory. Revised 9/98 Page 2

3 Let a and b be positive integers. The positive number c is called the least common multiple between a and b, denoted [a, b] or l.c.m.(a, b), if: a) a c and b c (c is a multiple of a and b) b) If a k and b k for a positive integer k then c k. (any common multiple of a and b is a multiple of c) In other words, the minimum common multiple is the smallest positive integers which is a common multiple between a and b. There is a way to find the least common multiple of two integers by using the prime number decomposition of them. The algorithm to do it is illustrated for the numbers 4 and 6: Find the prime number decomposition of 4 and 6 4 = 2 2 and 6 = Rewrite those products in such a way that the same prime numbers appear in each of the products: 4 = = Use the largest exponent of each of the prime numbers participating as factors in at least one to the products, to generate the least common multiple l.c.m.( 4, 6 ) = 2 MAX {2, 1} MAX { 0, 1} 3 = Thus, the least common multiple of 4 and 6 is = 12. EXERCISE 2 1. Calculate l.c.m. ( 50, 72) 2. In this case you want to analyze what happens with the lcm between two number when is a multiple of the other. a. What is the relationship between the set of multiples of 4 and the set of multiples of 8? b. What can you say about the l.c.m.(4, 8)? c. Do the answers to (a) and (b) work for any two integers whenever one number is a multiple of the other? Explain. GREATEST COMMON DIVISOR Now, we want to find the common divisors of 30 and 12. Let s start by considering the positive divisors of 30 and the positive divisors of 12. Call these sets M and N. M = {1, 2, 3, 5, 6, 10, 15, 30} N= {1, 2, 3, 4, 6, 12} Discrete Math. Number Theory. Revised 1/99 Page 3

4 The common divisors are M N = {2, 3, 6}. The greatest value of this set is 6 which is called the greatest common divisor. Two things to observe: 6 is a divisor of 12 and 30 any other common divisor of 12 and 30 divides 6. These observations lead to the formal definition that follows: Let a and b be positive integers. The positive integer c is the greatest common divisor between a and b, denoted g.c.d.(a, b) or (a, b), if: a) c a and c b (a and b are multiples of c) b) If k a and k b then k c. (Any divisor of a and b divides c as well.) In other words, the greatest common divisor it is the greatest positive integral number that divides a and b. As with the l.c.m. there is an algorithm to find the g.c.d. of two numbers. Let s follow the algorithm to find g.c.d (12, 30): Find the prime number decomposition of each number 12 = = Rewrite each of the products in such a way that the same factors participate in each of them 12 = = Produce a number with all the prime factors participating in the products but selecting the smallest exponent between two powers of the same prime number. g.c.d.( 12, 30) = 2 Min {2, 1} 3 Min {1, 1} Min {0,1} 5 = = 6 When the greatest common divisor between two numbers is 1, the numbers are called relatively prime. For example, 4 and 5, 10 and 21, are pairs of relatively prime numbers. But 8 and 14 are not. There is a way to relate the product of two positive integer numbers with their l.c.m and g.c.d. This is given by: For any two positive integers a and b, a b = [a, b] (a, b) EXERCISE 3 1. Give three pairs of numbers, which are not relative primes. Verify the previous result for each pair of numbers. 2. Give a pair of relatively prime numbers. Verify the previous result for each pair. 3. Using the particular cases in parts (1) and (2), justify why the result, to express the product of two integers as the product of their l.c.m. and their g.c.d. follows. (Hint: Consider the algorithms to calculate the l.c.m. and g.c.d.) Discrete Math. Number Theory. Revised 9/98 Page 4

5 4. Use the concept of least common multiple to verify each of the claims below. Explain your answers: a. Any multiple of 2 and 5 is a multiple of 10. b. Any multiple of 4 and 6 is a multiple of We know that 12 is a multiple of 4. What can you say about the g.c.d.(4, 12)? 6. What condition must satisfy the integers a and b to be able to guarantee that any multiple of a and b is also a multiple of ab? Give some examples to see what happens. THE DIVISION ALGORITHM Let s consider the multiples of 7 on a number line. 7 (-2) 7 (-1) 7 (0) 7 (1) 7 (2) 7 (3) Take any two consecutive multiples of 7, say 7 2 and 7 3. The numbers between them, including the first one but not the second, can be written as , , , , , , Hence, any integer that we take is a multiple of seven or between two consecutive multiples of seven. Therefore, it can be written as 7q + r, where r = 0, 1, 2, 3, 4, 5, 6 For example the number 82 can be written as , and the number -22 can be written as -22 = 7 (-4) + 6. Notice that q is selected so that 7q is the largest multiple of 7 less than or equal to the given number. It guarantees that r is always non-negative number less than or equal to 6. The generalization of this result is THE DIVISION ALGORITHM: Let a be an integer and d a positive integer. There are unique integers q and r such that a = dq + r, 0 r d - 1 In this equality a is called the dividend, q is the quotient and r the remainder. REMARK Using the division algorithm, we can characterize the even numbers as those integers with remainder zero when divided by two, and the odd numbers as those numbers with remainder one when divided by two. Discrete Math. Number Theory. Revised 1/99 Page 5

6 EXERCISE 4 1. Find the quotient and remainder of each of the following numbers when they are divided by 11: 0, -44, Below each of the given sets of 8 consecutive integer numbers write the remainder when they are divided by 3. After that, determine how many numbers in each set are multiples of 3. It is nice to observe the strings formed by the remainders. Is there any pattern? 2, 3, 4, 5, 6, 7, 8, 9 13, 14, 15, 16, 17, 18, 19, 20-18, -17, -16, -15, -14, -13, -12, Every time you have three consecutive integers there is exactly one which is multiple of 3. Why? 4. In a set of n consecutive integer how many are multiples of n? Explain 5. If you have eight consecutive integers, what conjecture can you make with respect to how many of them are multiples of 3? Explain. 6. Consider the integer numbers x such that 100 x < 200 a. How many are divisible by 4? b. How many are divisible by 6? c. How many are divisible by [4, 6]? MODULAR ARITHMETIC It is 12 noon right now. We want to determine which number the hour handle will be point to after 7 hours? 19 hours? 40 hours? 64 hours? It is not a coincidence that after 7 and 19 hours it is pointing to the same number. Likewise after 40 and 64 hours it points to the same number. Observe that 19-7 and are multiples of 12 (The hour hand is at the same place every twelve hours). We say that 7 and 19 are congruent module 12 and that 64 and 40 are congruent module 12 because their differences are multiples of 12. This is a formal definition: Let a and b be integers and m a positive integer. We say that a and b are congruent modulo m, or a mod m is b if a-b is a multiple of m. Observe that if a-b is a multiple of m then b-a is a multiple of m as well. That is if a mod m is b then b mod m is a. EXERCISE 5 1. Which of the following integers is (are) congruent to 6 modulo 18? Justify your answer Fill in the blank with the least non-negative integer number Discrete Math. Number Theory. Revised 9/98 Page 6

7 12 mod 5 is 80 mod 2 is -27 mod 8 is 3. Find three positive numbers and three negative numbers congruent to 8 modulo 13. OBSERVATION We started the discussion about congruence by noting that 19 was 7 modulo 6 because their difference was a multiple of 6. Observe that 19 and 7 have the same remainder when they are divided by 6, it is 1. This is another way of defining congruence: For any two integer a and b, if they have the same remainder when divided by a positive number m, they are said to be congruent modulo m, or simply that a is b modulo m. Since 29 and -7 have the same remainder when divided by 6, we have that 29 modulo 6 is 7. HOW TO CALCULATE MODULO USING THE CALCULATOR. You can use the calculator to find the value of a number modulo another number. For example, let s say we want to find 38,507modulo 6. To use the calculator divide 38, 507 by 6 and take the integer part of it. In this case it is 6,417. The number 38,507-6,417*6=5 is 38,507 modulo 6. EXERCISE 6 Verify all the results from exercise 5 using the latest definition of congruence modulo m. APPLICATIONS HASHING FUNCTIONS You are responsible for storing and processing the customer records for a large company. Each customer has an identification number (SSN). This number is called its key. The idea is to create linked lists with the same number (if possible) of linked records. Suppose the company has approximately 20,000 customers and the company s computer is capable of searching a list of 100 items in an acceptable amount of time. If the numbers assigned to the customers are evenly distributed (that will rarely happened!), there will be 200 lists with 100 entries each one. Let s create 201 lists by assigning values to each customer from the {0, 1,, 200}. These are the remainders when a number is divided by 201. Therefore to each key will be assigned the least non-negative number modulo 201. For instance, the customer with key will be assigned the number mod 201 which is 129. It means that this record will be assigned to the list 129. Discrete Math. Number Theory. Revised 1/99 Page 7

8 However, there may be customers, which are assigned to the same list. For example the customer with key will be assigned to the same list. We say that a collision has occurred. In this case one thing that can be done is to insert the new record at the end of the existing list. There are other methods to make this assignment, but it is the simplest one. Discrete Math. Number Theory. Revised 9/98 Page 8