Math 110 FOUNDATIONS OF THE REAL NUMBER SYSTEM FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS

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1 4-1Divisibility Divisibility Divisibility Rules Divisibility An integer is if it has a remainder of 0 when divided by 2; it is otherwise. We say that 3 divides 18, written, because the remainder is 0 when 18 is divided by 3. Likewise, b divides a can be written. We say that 3 does not divide 25, written, because the remainder is not 0 when 25 is divided by 3. In general, if a is a nonnegative integer and b is a positive integer, we say that, or if and only if the remainder is 0 when a is divided by b. Definition If a and b are any integers, then b divides a, written b a, if, and only if,. If b a, then b is a factor or a divisor of a, and a is a multiple of b. Example 1 Classify each of the following as true or false. a b. 0 5 c. 0 is odd d. For all integers a, 1 a. e. For all integers a, 1 a. f. 3 9n for all integers n. g. h Divisibility Page 1

2 Properties of Division For any integers a and d, if d a, and n is any integer, then d na. In other words, if d is a factor of a, then d is a factor of any multiple of a. For any integers a, b, and d, d 0, a. If d a, and d b, then d (a + b). b. if d a, and, then c. If d a, and d b, then d (a b). d. If d a, and, then Example 2 Classify each of the following as true or false, where x, y, and z are integers. a. If 5 x and 5 y, then 5 xy. b. If 5 (x + y), then 5 x and 5 y. c. Divisibility Rules Sometimes it is useful to know if one number is divisible by another just by looking at it. For example, to check the divisibility of 1734 by 17, we note that 1734 = We know that because and 17 divides any multiple of 17. Furthermore, 17 34; therefore, we conclude that Another method to check for divisibility is to use the integer division button a calculator. on Press the following sequence of buttons: What is your answer? 4.1 Divisibility Page 2

3 Divisibility Tests Divisible by: 2 If and only if Examples Example 3 a. Determine whether 37,256 is divisible by 2, 4, and 8. b. Determine whether 57,034 is divisible by 2, 4, and Divisibility Page 3

4 Example 4 a. Determine whether 1005 is divisible by 3 and 9. b. Determine whether 13,428 is divisible by 3 and 9. Example 5 The store manager has an invoice for 72 four-function calculators. The first and last digits on the receipt are illegible. The manager can read $ 67.9 What are the missing digits, and what is the cost of each calculator? Example 6 The number 57,729,364,583 has too many digits for most calculator displays. Determine whether it is divisible by each of the following: a. 2 b. 3 c. 5 d. 6 e. 8 f. 9 g. 10 h Divisibility Page 4

5 Homework 1. Place all possible digits in the square, so that the number 527,4 2 is divisible by a. 2 b. 3 c. 4 d. 6 e. 9 f Classify each of the following as true of false. If false, tell why. a. 5 is multiple of 0. b. 10 is a divisor of 30. c d. 10 is divisible by 1 e. 30 is a factor of 6 f. 6 is a multiple of A student writes, If d a and d b, then d (a + b). How do you respond? 4.1 Divisibility Page 5

6 4-2Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisors Determining if a Number is Prime More About Primes Prime and Composite Numbers Prime number Any positive integer with exactly two distinct, positive divisors Composite number Any integer greater than 1 that has a positive factor other than 1 and itself Example 1 Show that the following numbers are composite. a b c d Fundamental Theorem of Arithmetic (Unique Factorization Theorem) Each composite number can be written as a product of primes in one and only one way except for the order of the prime factors in the product. Prime Factorization A factorization containing only prime numbers is a. To find the prime factorization of a composite number, we first rewrite the number as a product of two smaller natural numbers greater than 1. We continue the process, factoring the lesser numbers until all factors are prime. Factor tree Ex: Find the prime factorization of Prime and Composite Number Page 1

7 Ex: Find the prime factorization of 260 Dividing primes We can also determine the prime factorization by dividing with the least prime, 2, if possible. If not, we try the next larger prime as a divisor. Once we find a prime that divides the number, we continue by finding smallest prime that divides that quotient, etc. Ex: Find the prime factorization of 260 Number of Divisors How many positive divisors does 24 have? We are not asking how many prime divisors, just the number of divisors any divisors. Let the prime factorization of n= p a q b r c... then the number of positive divisors, d(n) = (a+1)(b+1)(c+1)... Ex: Find the number of positive divisor of Prime and Composite Number Page 2

8 Example 2 a. Find the number of positive divisors of 1,000,000. b. Find the number of positive divisors of Determining if a Number is Prime To determine if a number is prime, you must check only divisibility by prime numbers less than the given number. For example, to determine if 97 is prime, we must try dividing 97 by the prime numbers: 2, 3, 5, and so on as long as the prime is less than 97. Theorem 4-14 If d is divisor of n, then n d is also a divisor of n. Theorem 4.15 If n is composite, then n has a prime factor p such that p 2 n. 4.2 Prime and Composite Number Page 3

9 Theorem 4-16 If n is a whole number greater than 1 and not divisible by any prime p such that p 2 n, then n is prime. Example 3 a. Is 397 composite or prime? b. Is 91 composite or prime? Sieve of Eratosthenes One way to find all the primes less than a given number is to use the Sieve of Eratosthenes. If all the natural numbers greater than 1 are considered (or placed in the sieve), the numbers that are not prime are methodically crossed out (or drop through the holes of the sieve). The remaining numbers are prime. Q1. List the prime numbers from 1 to 100. Q2: Explain why some numbers could be crossed out more than once? 4.2 Prime and Composite Number Page 4

10 Homework 1. (Problem Solving)A large toy store carries one kind of stuffed bear. On Monday the store sold a certain number of the stuffed bears for a total of $1843 and on Tuesday, without changing the price, the store sold a certain number of the stuffed bears for a total of $1957. How many toy bears were sold each day if the price of each bear is a whole number and greater than $1? 2. In order to test for divisibility by 12, one student checked to determine divisibility by 3 and 4; another checked for divisibility by 2 and 6. Are both students using a correct approach to divisibility by 12? Why or why not? 4.2 Prime and Composite Number Page 5

11 4-3 Greatest Common Divisor and Least Common Multiple Methods to Find the Greatest Common Divisor Methods to Find the Least Common Multiple Greatest Common Divisor Two bands are to be combined to march in a parade. A 12- member band will march behind a 18-member band. The combined bands must have the same number of columns. Each column must be the same size. What is the greatest number of columns in which they can march? Greatest Common Divisor (GCD) The greatest common divisor (GCD) of two natural numbers a and b is the greatest natural number that divides both a and b. Colored Rods Method (Cuisenaire Rods) Find the GCD of 6 and 8 using the 6 rod and the 8 rod. Find the longest rod such that we can use multiples of that rod to build both the 6 rod and the 8 rod. The 2 rods can be used to build both the 6 and 8 rods. The 3 rods can be used to build the 6 rod but not the 8 rod. The 4 rods can be used to build the 8 rod but not the 6 rod. The 5 rods can be used to build neither. The 6 rods cannot be used to build the 8 rod. Therefore, GCD(6, 8) = Greatest Common Divisor and Least Common Multiple Page 1

12 The Intersection-of-Sets Method List all members of the set of positive divisors of the two integers, then find the set of all common divisors, and, finally, pick the greatest element in that set. To find the GCD of 20 and 32, denote the sets of divisors of 20 and 32 by D20 and D32, respectively. Because the greatest number in the set of common positive divisors is 4, GCD(20, 32) = 4. The Prime Factorization Method To find the GCD of two or more positive integers, first find the prime factorizations of the given numbers and then identify each common prime factor of the given numbers. The GCD is the product of the common factors, each raised to the lowest power of that prime that occurs in any of the prime factorizations. Numbers, such as 4 and 9, whose GCD is 1 are relatively prime. Example 1 Find each of the following: a. GCD(108, 72) b. GCD(4, 9) c. GCD(x, y) if x = and y = d. GCD(x, y, z) if x = , y = , and z = e. GCD(x, y) if x = and y = Greatest Common Divisor and Least Common Multiple Page 2

13 Euclidean Algorithm Method If a and b are any whole numbers greater than 0 and a b, then GCD(a, b) = GCD(r, b), where r is the remainder when a is divided by b. Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is 0 is called the Euclidean algorithm. Example 2 Use the Euclidean algorithm to find GCD(10764, 2300). Example 3 a. Find GCD(134791, 6341, 6339). b. Find the GCD of any two consecutive whole numbers. 4.3 Greatest Common Divisor and Least Common Multiple Page 3

14 Least Common Multiple (LCM) Hot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a package. What is the least number of packages of each you must buy so that there is an equal number of hot dogs and buns? Suppose that a and b are positive integers. Then the least common multiple (LCM) of a and b is the least positive integer that is simultaneously a multiple of a and a multiple of b. Number-Line Method Find LCM(3, 4). Beginning at 0, the arrows do not coincide until the point 12 on the number line. Thus, 12 is LCM(3, 4). Colored Rods Method(Cuisenaire Rods) Find LCM(3, 4) using the 3 rod and the 4 rod. Build trains of 3 rods and 4 rods until they are the same length. The LCM is the common length of the train. 4.3 Greatest Common Divisor and Least Common Multiple Page 4

15 The Intersection-of-Sets Method List all members of the set of positive multiples of the two integers, then find the set of all common multiples, and, finally, pick the least element in that set. To find the LCM of 8 and 12, denote the sets of positive multiple of 8 and 12 by M8 and M12, respectively. Because the least number in he set of common positive multiples is 24, LCM(8, 12) = 24. Because the least number in he set of common positive multiples is 24, LCM(8, 12) = 24. The Prime Factorization Method To find the LCM of two natural numbers, first find the prime factorization of each number. Then take each of the primes that are factors of either of the given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations. Example 4 Find LCM(2520, 10530). GCD-LCM Product Method For any two natural numbers a and b, we have GCD(a, b) LCM(a, b) = ab. Example 5 Find LCM(731, 952). 4.3 Greatest Common Divisor and Least Common Multiple Page 5

16 Division-by-Primes Method To find LCM(12, 75, 120), start with the least prime that divides at least one of the given numbers. Divide as follows: Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this procedure, continue the division process until the row of answers consists of relatively prime numbers as shown next. Homework 1. Find the GCD and the LCM for each of the following using the intersection-of-sets method: a. 12 and 18 b. 18 and 36 c. 12, 18 and 24 d. 6 and The principle of Valley Elementary School wants to divide each of the three fourthgrade classes into small same-size groups with at least 2 students in each. If the classes have 18, 24, and 36 students, respectively, what size groups are possible? 4.3 Greatest Common Divisor and Least Common Multiple Page 6

17 5-1Integers and the Operations of Addition and Subtraction Representations of Integers Integer Addition Number-Line Model Absolute Value Properties of Integer Addition Integer Subtraction Representations of Integers The set of integers is denoted by I: The negative integers are opposites of the positive integers. 4 is the opposite of positive 3 is the opposite of 3 Example 1 For each of the following, find the opposite of x. a. x = 3 b. x = 5 c. x = 0 Integer Addition Chip Model for Addition Black chips represent positive integers and red chips represent negative integers. Each pair of black/red chips neutralize each other. 5.1 Integers and the Operations of Addition and Subtraction Page 1

18 Charged-Field Model Similar to the chip model. Positive integers are represented by + s and negative integers by s. Positive charges neutralize negative charges. Number-Line Model Positive integers are represented by moving forward (right) on the number line; negative integers are represented by moving backward (left). Example 2 The temperature was 4 C. In an hour, it rose 10 C. What is the new temperature? Pattern Model Beginning with whole number facts, a table of computations is created by following a pattern. 5.1 Integers and the Operations of Addition and Subtraction Page 2

19 Absolute Value The absolute value of a number a, written a, is the distance on the number line from 0 to a. Definition Example 3 Evaluate each of the following. a. 20 b. 5 c. 0 d. 3 e Properties of Integer Addition Integer addition has all the properties of whole- number addition. 5.1 Integers and the Operations of Addition and Subtraction Page 3

20 Properties of the Additive Inverse By definition, the additive inverse, a, is the solution of the equation x + a = 0. For any integers a and b, the equation x + a = b has a unique solution, b + a. Example 4 Find the additive inverse of each of the following. a. (3 + x) b. a + 4 c. 3 + x Integer Subtraction Chip Model for Subtraction To find 3 2, add 0 in the form (two black chips and two red chips) to the three black chips, then take away the two red chips. 5.1 Integers and the Operations of Addition and Subtraction Page 4

21 Charged-Field Model for Subtraction To find 3 5, represent 3 so that at least five negative charges are present. Then take away the five negative charges. Number-Line Model While integer addition is modeled by maintaining the same direction and moving forward or backward depending on whether a positive or negative integer is added, subtraction is modeled by turning around. 5.1 Integers and the Operations of Addition and Subtraction Page 5

22 Pattern Model for Subtraction Using inductive reasoning and starting with known subtraction facts, find the difference of two integers by following a pattern. Subtraction Using the Missing Addend Approach Subtraction of integers, like subtraction of whole numbers, can be defined in terms of addition. 5.1 Integers and the Operations of Addition and Subtraction Page 6

23 Definition of Integer Subtraction For integers a and b, a b is the unique integer n such that a = b + n. Example 5 Use the definition of subtraction to compute the following: a b Subtraction Using Adding the Opposite Approach Subtracting an integer is the same as adding its opposite. Example 6 Using the fact that a b = a + b, compute each of the following: a. 2 8 b. 2-8 c d Example 7 Write expressions equal to each of the following without parentheses. a. (b c) b. a (b + c) Example 8 Simplify each of the following. a. 2 (5 x) b. 5 (x 3) c. (x y) y 5.1 Integers and the Operations of Addition and Subtraction Page 7

24 Order of Operations Recall that subtraction is neither commutative nor associative. An expression such as is ambiguous unless we know in which order to perform the subtractions. Mathematicians agree that means (3 15) 8. Subtractions are performed in order from left to right. Example 9 Compute each of the following. a b c. 3 (7 3) Homework 1. I am choosing an integer. I then subtract 10 from the integer, take the opposite of the result, add -3, and find the opposite of the new result. My result is -3, and finds the opposite of the new result. My result is -3. What is the original number? 2. A fourth-grade student devised the following algorithm for subtracting 84-27: 4 minus 7 equals negative minus 20 equals plus negative 3 equals Thus the answer is 57. How should you respond as a teacher? Will this technique always work? 5.1 Integers and the Operations of Addition and Subtraction Page 8

25 5-2MultiplicationandDivisionofIntegers Models Properties of Integer Multiplication Integer Division Order of Operations on Integers Ordering Integers Integer Multiplication Models Pattern Model First find (3)( 2) using repeated addition: (3)( 2) = = 6 Now use the commutative property to find ( 2)(3)= Chip Model Charged-Field Model Chip Model To find ( 3)( 2) = 6, start with a value of 0 that includes at least 6 red chips, then remove 6 red chips. 5.2 Multiplication and Division of Integers Page 1

26 Charged-Field Model Number-Line Model Demonstrate multiplication by using a hiker moving along a number line. Traveling to the left (west) means moving in the negative direction, and traveling to the right (east) means moving in the positive direction. Time in the future is denoted by a positive value, and time in the past is denoted by a negative value. 5.2 Multiplication and Division of Integers Page 2

27 Properties of Integer Multiplication For all integers a, b, c I, the set of integers: For all integers a, b, and c, 1. ( 1)a = a. 2. ( a)b = b( a) = (ab). 3. ( a)( b) = ab. Distributive property of multiplication over subtraction for integers: (b c) = ab ac and (b c)a = ba ca 5.2 Multiplication and Division of Integers Page 3

28 Example 1 Simplify each of the following so that there are no parentheses in the final answer: a. 3(x 2) b. (a + b)(a b) Example 2 Use the difference-of-squares formula to simplify the following: a. (4 + b)(4 b) b. ( 4 + b)( 4 b) Factoring When the distributive property of multiplication over subtraction is written in reverse order as ab ac = a(b c) and ba ca = (b c)a and similarly for addition, the expressions on the right of each equation are in factored form. Both the difference-of-squares formula and the distributive properties of multiplication over addition and subtraction can be used for factoring. Example 3 Factor each of the following completely: a. x 2 9 b. (x + y) 2 z 2 c. -3x + 5xy d. 3x Multiplication and Division of Integers Page 4

29 Integer Division Definition of Integer Division If a and b are any integers, then a b is the unique integer c, if it exists, such that a = bc. The quotient of two negative integers, if it exists, is a positive integer. The quotient of a positive and a negative integer, if it exists, is a negative integer. Example 4 Use the definition of integer division, if possible, to evaluate each of the following: a. 12 ( 4) b c. 12 ( 4) d e. (ab) b, b 0 f. (ab) a, a 0 Order of Operations on Integers When addition, subtraction, multiplication, division, and exponentiation appear without parentheses: 1. Exponentiation is done first. 2. Multiplication and division in the order of their appearance from left to right. 3. Finally, addition and subtraction in the order of their appearance from left to right. Arithmetic operations inside parentheses must be done first according to rules Multiplication and Division of Integers Page 5

30 Example 5 Evaluate each of the following: a b. (2 5)4 + 1 c d e. (-3) 4 f Ordering Integers Definition of Less than for Integers For any integers a and b, a is less than b, written a < b, if, and only if, there exists a positive integer k such that a + k = b. a < b (or equivalently, b > a) if, and only if, b a is equal to a positive integer; that is, b a is greater than 0. Properties of Inequalities of Integers 1. If x < y, and n is any integer, then x + n < y + n. 2. If x < y, then x > y. 3. If x < y and n > 0, then nx < ny. 4. If x < y and n < 0, then nx > ny. 5.2 Multiplication and Division of Integers Page 6

31 Example 6 Find all integers x that satisfy the following: a. x + 3 < 2 b. x 3 < 5 Homework 1. A 7 th grader student does not believe that -5 < 2. The student argues that a debt of $5 is greater than a debt of $2. How do you respond? 2. A student computes 8 2( 3) by writing 10( 3) = 30. How would your help this student? 3. Betty used the charged field model to show that 2( 3) = 6. She said that this proves that any negative integer times a negative integer is a positive integer. How do you respond? 5.2 Multiplication and Division of Integers Page 7

32 4-1Divisibility Divisibility Divisibility Rules Divisibility An integer is even if it has a remainder of 0 when divided by 2; it is odd otherwise. We say that 3 divides 18, written 3 18, because the remainder is 0 when 18 is divided by 3. Likewise, b divides a can be written b a. We say that 3 does not divide 25, written, because the remainder is not 0 when 25 is divided by 3. In general, if a is a nonnegative integer and b is a positive integer, we say that a is divisible by b, or b divides a if and only if the remainder is 0 when a is divided by b. Definition If a and b are any integers, then b divides a, written b a, if, and only if, there is a unique integer q such that a = bq. If b a, then b is a factor or a divisor of a, and a is a multiple of b. Example 1 Classify each of the following as true or false. a b. 0 5 c. 0 is odd d. For all integers a, 1 a. e. For all integers a, 1 a. f. 3 9n for all integers n. g. h Divisibility Page 1

33 Properties of Division For any integers a and d, if d a, and n is any integer, then d na. In other words, if d is a factor of a, then d is a factor of any multiple of a. For any integers a, b, and d, d 0, a. If d a, and d b, then d (a + b). b. if d a, and, then c. If d a, and d b, then d (a b). d. If d a, and, then Example 2 Classify each of the following as true or false, where x, y, and z are integers. a. If 5 x and 5 y, then 5 xy. b. If 5 (x + y), then 5 x and 5 y. c. Divisibility Rules Sometimes it is useful to know if one number is divisible by another just by looking at it. For example, to check the divisibility of 1734 by 17, we note that 1734 = We know that because and 17 divides any multiple of 17. Furthermore, 17 34; therefore, we conclude that Another method to check for divisibility is to use the integer division button a calculator. on Press the following sequence of buttons: What is your answer? 4.1 Divisibility Page 2

34 Divisibility Tests An integer is divisible by 2 if and only if its units digit is divisible by 2. An integer is divisible by 5 if and only if its units digit is divisible by 5, that is if and only if the units digit is 0 or 5. An integer is divisible by 10 if and only if its units digit is divisible by 10, that is if and only if the units digit is 0. An integer is divisible by 4 if and only if the last two digits of the integer represent a number divisible by 4. An integer is divisible by 8 if and only if the last three digits of the integer represent a number divisible by 8. Example 3 a. Determine whether 37,256 is divisible by 2, 4, and 8. b. Determine whether 57,034 is divisible by 2, 4, and 8. Divisibility Tests Divisibility Tests An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. An integer is divisible by 9 if and only if the sum of the digits of the integer is divisible by Divisibility Page 3

35 Example 4 a. Determine whether 1005 is divisible by 3 and 9. b. Determine whether 13,428 is divisible by 3 and 9. Example 5 The store manager has an invoice for 72 four-function calculators. The first and last digits on the receipt are illegible. The manager can read $ 67.9 What are the missing digits, and what is the cost of each calculator? Divisibility Tests for 11 and 6 An integer is divisible by 11 if and only if the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11. An integer is divisible by 6 if and only if the integer is divisible by both 2 and Divisibility Page 4

36 Example 6 The number 57,729,364,583 has too many digits for most calculator displays. Determine whether it is divisible by each of the following: a. 2 b. 3 c. 5 d. 6 e. 8 f. 9 g. 10 h. 11 Homework 1. Place all possible digits in the square, so that the number 527,4 2 is divisible by a. 2 b. 3 c. 4 d. 6 e. 9 f Classify each of the following as true of false. If false, tell why. a. 5 is multiple of 0. b. 10 is a divisor of 30. c d. 10 is divisible by 1 e. 30 is a factor of 6 f. 6 is a multiple of Divisibility Page 5

37 5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisors Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognize that different types of numbers have particular characteristics; for example, square numbers have an odd number of factors and prime numbers have only two factors. The following rectangles represent the number 18. The number 18 has 6 positive divisors: 1, 2, 3, 6, 9 and 18. Number of Positive Divisors Prime and Composite Numbers Below each number listed across the top, we identify numbers less than or equal to 37 that have that number of positive divisors.

38 Prime number Any positive integer with exactly two distinct, positive divisors Composite number Any integer greater than 1 that has a positive factor other than 1 and itself Example 1 Show that the following numbers are composite. a b c d Prime Factorization Students continue to develop their understanding of multiplication and division and the structure of numbers by determining if a counting number greater than 1 is a prime, and if it is not, by factoring it into a product of primes. Composite numbers can be expressed as products of two or more whole numbers greater than 1. Each expression of a number as a product of factors is a factorization. A factorization containing only prime numbers is a prime factorization. Fundamental Theorem of Arithmetic (Unique Factorization Theorem) Each composite number can be written as a product of primes in one and only one way except for the order of the prime factors in the product.

39 Prime Factorization To find the prime factorization of a composite number, rewrite the number as a product of two smaller natural numbers. If these smaller numbers are both prime, you are finished. f either is not prime, then rewrite it as the product of smaller natural numbers. Continue until all the factors are prime. The two trees produce the same prime factorization, except for the order in which the primes appear in the products. Prime Factorization We can also determine the prime factorization by dividing with the least prime, 2, if possible. If not, we try the next larger prime as a divisor. Once we find a prime that divides the number, we continue by finding smallest prime that divides that quotient, etc.

40 Number of Divisors How many positive divisors does 24 have? We are not asking how many prime divisors, just the number of divisors any divisors. Since 1 is a divisor of 24, then 24/1 = 24 is a divisor of 24. Since 2 is a divisor of 24, then 24/2 = 12 is a divisor of 24. Since 3 is a divisor of 24, then 24/3 = 8 is a divisor of 24. Since 4 is a divisor of 24, then 24/4 = 6 is a divisor of 24. Another way to think of the number of positive divisors of 24 is to consider the prime factorization 2 3 = 8 has four divisors. 3 has two divisors. Using the Fundamental Counting Principle, there are 4 2 = 8 divisors of 24.

41 If p and q are different primes, then p n q m has (n + 1)(m + 1) positive divisors. In general, if p 1, p 2,, p k are primes, and n 1, n 2,, n k are whole numbers, then has positive divisors. Example 2 a. Find the number of positive divisors of 1,000,000. b. Find the number of positive divisors of Determining if a Number is Prime To determine if a number is prime, you must check only divisibility by prime numbers less than the given number. For example, to determine if 97 is prime, we must try dividing 97 by the prime numbers: 2, 3, 5, and so on as long as the prime is less than 97. If none of these prime numbers divide 97, then 97 is prime. Upon checking, we determine that 2, 3, 5, 7 do not divide 97. Assume that p is a prime greater than 7 and p 97. Then 97/p also divides 97. Because p 11, then 97/p must be less than 10 and hence cannot divide 97.

42 If d is a divisor of n, then is also a divisor of n. If n is composite, then n has a prime factor p such that p 2 n. If n is an integer greater than 1 and not divisible by any prime p, such that p 2 n, then n is prime. Note: Because p 2 n implies that than or equal to is a divisor of n. it is enough to check if any prime less Example 3 a. Is 397 composite or prime? b. Is 91 composite or prime? Sieve of Eratosthenes One way to find all the primes less than a given number is to use the Sieve of Eratosthenes. If all the natural numbers greater than 1 are considered (or placed in the sieve), the numbers that are not prime are methodically crossed out (or drop through the holes of the sieve). The remaining numbers are prime. 5.4 Homework # A-2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19

43 5-5 GreatestCommonDivisorandLeastCommonMultiple Methods to Find the Greatest Common Divisor Methods to Find the Least Common Multiple Greatest Common Divisor Two bands are to be combined to march in a parade. A 24-member band will march behind a 30-member band. The combined bands must have the same number of columns. Each column must be the same size. What is the greatest number of columns in which they can march? The bands could each march in 2 columns, and we would have the same number of columns, but this does not satisfy the condition of having the greatest number of columns. The number of columns must divide both 24 and 30. Numbers that divide both 24 and 30 are 1, 2, 3, and 6. The greatest of these numbers is 6. The first band would have 6 columns with 4 members in each column, and the second band would have 6 columns with 5 members in each column. Greatest Common Divisor (GCD) The greatest common divisor (GCD) of two natural numbers a and b is the greatest natural number that divides both a and b. Colored Rods Method Find the GCD of 6 and 8 using the 6 rod and the 8 rod. Find the longest rod such that we can use multiples of that rod to build both the 6 rod and the 8 rod. The 2 rods can be used to build both the 6 and 8 rods. The 3 rods can be used to build the 6 rod but not the 8 rod. The 4 rods can be used to build the 8 rod but not the 6 rod. The 5 rods can be used to build 4.3 Greatest Common Divisor and Least Common Multiple Page 1

44 neither. The 6 rods cannot be used to build the 8 rod. Therefore, GCD(6, 8) = 2. The Intersection-of-Sets Method List all members of the set of positive divisors of the two integers, then find the set of all common divisors, and, finally, pick the greatest element in that set. To find the GCD of 20 and 32, denote the sets of divisors of 20 and 32 by D 20 and D 32, respectively. Because the greatest number in the set of common positive divisors is 4, GCD(20, 32) = 4. The Prime Factorization Method To find the GCD of two or more positive integers, first find the prime factorizations of the given numbers and then identify each common prime factor of the given numbers. The GCD is the product of the common factors, each raised to the lowest power of that prime that occurs in any of the prime factorizations. Numbers, such as 4 and 9, whose GCD is 1 are relatively prime. Example 1 Find each of the following: a. GCD(108, 72) b. GCD(0, 13) c. GCD(x, y) if x = and y = d. GCD(x, y, z) if x = , y = , and z = Greatest Common Divisor and Least Common Multiple Page 2

45 e. GCD(x, y) if x = and y = Calculator Method Try finding the Greatest Common Divisor on a calculator. Euclidean Algorithm Method If a and b are any whole numbers greater than 0 and a b, then GCD(a, b) = GCD(r, b), where r is the remainder when a is divided by b. Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is 0 is called the Euclidean algorithm. Example 2 Use the Euclidean algorithm to find GCD(10764, 2300). Example 3 a. Find GCD(134791, 6341, 6339). b. Find the GCD of any two consecutive whole numbers. 4.3 Greatest Common Divisor and Least Common Multiple Page 3

46 Least Common Multiple Hot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a package. What is the least number of packages of each you must buy so that there is an equal number of hot dogs and buns? Least Common Multiple (LCM) Suppose that a and b are positive integers. Then the least common multiple (LCM) of a and b is the least positive integer that is simultaneously a multiple of a and a multiple of b. Number-Line Method Find LCM(3, 4). Beginning at 0, the arrows do not coincide until the point 12 on the number line. Thus, 12 is LCM(3, 4). Colored Rods Method Find LCM(3, 4) using the 3 rod and the 4 rod. Build trains of 3 rods and 4 rods until they are the same length. The LCM is the common length of the train. 4.3 Greatest Common Divisor and Least Common Multiple Page 4

47 The Intersection-of-Sets Method List all members of the set of positive multiples of the two integers, then find the set of all common multiples, and, finally, pick the least element in that set. To find the LCM of 8 and 12, denote the sets of positive multiple of 8 and 12 by M 8 and M 12, respectively. Because the least number in he set of common positive multiples is 24, LCM(8, 12) = 24. Because the least number in he set of common positive multiples is 24, LCM(8, 12) = 24. The Prime Factorization Method To find the LCM of two natural numbers, first find the prime factorization of each number. Then take each of the primes that are factors of either of the given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations. Example 4 Find LCM(2520, 10530). GCD-LCM Product Method For any two natural numbers a and b, GCD(a, b) LCM(a, b) = ab. Example 5 Find LCM(731, 952). 4.3 Greatest Common Divisor and Least Common Multiple Page 5

48 Division-by-Primes Method To find LCM(12, 75, 120), start with the least prime that divides at least one of the given numbers. Divide as follows: Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this procedure, continue the division process until the row of answers consists of relatively prime numbers as shown next. Homework 1. Find the GCD and the LCM for each of the following using the intersection-of-sets method: a. 12 and 18 b. 18 and 36 c. 12, 18 and 24 d. 6 and The principle of Valley Elementary School wants to divide each of the three fourth-grade classes into small same-size groups with at least 2 students in each. If the classes have 18, 24, and 36 students, respectively, what size groups are possible? 4.3 Greatest Common Divisor and Least Common Multiple Page 6