Politics and the political process affect everyone in some way. In local, state or national elections, registered voters make decisions about who will represent them and make choices about various ballot measures. In major issues at the state and national levels, pollsters use mathematics (in particular, statistics and statistical methods) to indicate attitudes and to predict, within certain percentages, how the electorate will vote. When there is an important election in your area, read the papers and magazines and listen to the television reports for mathematically related statements predicting the outcome. Consider the following situation. There are 8000 registered voters in Brownsville, and 8 of these voters live in neighborhoods on the north-side of town. A survey indicates that north-side voters are in favor of a bond measure for constructing a new recreation facility that would largely benefit their neighborhoods. 7 Also, of the registered voters from all other parts of town are 0 in favor of the measure. We might then want to know the following numbers. a. How many north-side voters favor the bond measure? b. How many voters in the town favor the bond measure? of these For more problems like these, see Section., Exercise 99.
Fractions, Mixed Numbers, and Proportions. Tests for Divisibility. Prime Numbers. Prime Factorization. Least Common Multiple (LCM). Introduction to Fractions.6 Division with Fractions.7 Addition and Subtraction with Fractions.8 Introduction to Mixed Numbers.9 Multiplication and Division with Mixed Numbers.0 Addition and Subtraction with Mixed Numbers. Complex Fractions and Order of Operations. Solving Equations with Fractions. Ratios and Proportions Chapter : Index of Key Terms and Ideas Chapter : Test Cumulative Review: Chapters -
Objectives A B Know the rules for testing divisibility by,,, 6, 9, and 0. Be able to apply the concept of divisibility to products of whole numbers. Teaching Note: This chapter is designed for students to develop an understanding of factors and the related skills needed for operations with fractions. All the topics are an integral part of the development of fractions and mixed numbers in Chapters and. Many of these ideas carry over into our work with decimal numbers, percents, and simplification of algebraic expressions.. Tests for Divisibility Objective A Tests For Divisibility (,,, 6, 9, 0) In our work with factoring and fractions, we will need to be able to divide quickly and easily by small numbers. Since we will be looking for factors, we will want to know if a number is exactly divisible (remainder 0) by some number before actually dividing. There are simple tests that can be performed mentally to determine whether a number is divisible by,,, 6, 9, or 0 without actually dividing. For example, can you tell (without dividing) if 8 is divisible by? By? Note that we are not trying to find the quotient, only to determine whether or is a factor of 8. The answer is that 8 is not divisible by and is divisible by. 9 ) 8 8 8 0 remainder 9 ) 8 8 7 0 remainder Thus the number is not a factor of 8. However, 9 8, and and 9 are factors of 8. Tests for Divisibility of Integers by,,, 6, 9, and 0 For : If the last digit (units digit) of an integer is 0,,, 6, or 8, then the integer is divisible by. For : If the sum of the digits of an integer is divisible by, then the integer is divisible by. For : If the last digit of an integer is 0 or, then the integer is divisible by. For 6: If the integer is divisible by both and, then it is divisible by 6. For 9: If the sum of the digits of an integer is divisible by 9, then the integer is divisible by 9. For 0: If the last digit of an integer is 0, then the integer is divisible by 0. 6 Chapter Fractions, Mixed Numbers, and Proportions
There are other quick tests for divisibility by other numbers such as, 7, and 8. (See Exercise for divisibility by.) Even and Odd Integers Even integers are divisible by. (If an integer is divided by and the remainder is 0, then the integer is even.) Teaching Note: The test for divisibility by has been included as an exercise. You may or may not want to include this topic in your class discussions and exams. Odd integers are not divisible by. (If an integer is divided by and the remainder is, then the integer is odd.) Note: Every integer is either even or odd. The even integers are, 0, 8, 6,,, 0,,, 6, 8, 0, The odd integers are,, 9, 7,,,,,,, 7, 9,, If the units digit of an integer is one of the even digits (0,,, 6, 8), then the integer is divisible by, and therefore, it is an even integer. Example Testing Divisibility a. 86 is divisible by since the units digit is 6, an even digit. Thus 86 is an even integer. b. 770 is divisible by because 7 + 7 + 0 +, and is divisible by. c.,6 is divisible by because the units digit is.. Test the divisibility of the following examples. a. Does 6 divide 80? Explain why or why not. b. Does divide 06? Explain why or why not. d. 906 is divisible by 6 since it is divisible by both and. (The sum of the digits is 9 + 0 + + 6 8, and 8 is divisible by. The units digit is 6 so 906 is divisible by.) e. 967 is divisible by 9 because 9 + + 6 + 7 7 and 7 is divisible by 9. f.,0 is divisible by 0 because the units digit is 0. Now work margin exercise. c. Is 06 divisible by? By 9? d. Is,7 divisible by? By 0? a. Yes, divides the sum of the digits and the units digit is. b. No, does not divide the sum of the digits. c. yes; no d. yes; no Tests for Divisibility Section. 6
Note about Terminology The following sentences are simply different ways of saying the same thing.. 86 is divisible by.. is a factor of 86.. divides 86.. divides into 86.. is a divisor of 86. In examples and, all six tests for divisibility are used to determine which of the numbers,,, 6, 9, and 0 will divide into each number. Example Testing for Divisibility of 0 The number 0 is a. divisible by (units digit is 0, an even digit); b. not divisible by ( + + + 0 0 and 0 is not divisible by ); c. divisible by (units digit is 0); d. not divisible by 6 (to be divisible by 6, it must be divisible by both and, but 0 is not divisible by ); e. not divisible by 9 ( + + + 0 0 and 0 is not divisible by 9); f. divisible by 0 (units digit is 0). Example Testing for Divisibility of -78 The number 78 is a. divisible by (units digit is 8, an even digit); b. divisible by ( + 7 + + 8 and is divisible by ); c. not divisible by (units digit is not 0 or ); d. divisible by 6 (divisible by both and ); e. not divisible by 9 ( + 7 + + 8 and is not divisible by 9); f. not divisible by 0 (units digit is not 0). 6 Chapter Fractions, Mixed Numbers, and Proportions
Completion Example Testing for Divisibility of -7 The number 7 is a. divisible by because. b. divisible by because. c. not divisible by 6 because. Determine which of the numbers,,, 6, 9, and 0 divides into each of the following numbers.. 7. 800.,0. 00 Completion Example Testing for Divisibility of 6 The number 6 is a. divisible by 6 because...,, 0. none.,,, 6, and 0 b. divisible by 9 because. c. not divisible by 0 because. Now work margin exercises through. Objective B Divisibility of Products Now consider a number that is written as a product of several factors. For example, 0 700. To better understand how products and factors are related, consider the problem of finding the quotient 700 divided by 0. 700 0? Completion Example Answers. a. the units digit is. b. the sum of the digits is, and is divisible is. c. 7 is not even.. a. 6 is divisible by both and. b. the sum of the digits is 9, and 9 is divisible by 9. c. the units digit is not 0. Tests for Divisibility Section. 6
By using the commutative and associative properties of multiplication (i.e. rearranging and grouping factors), we can see that 0 0 and 700 ( 0) ( ) 0 8. Thus 700 0 8. In general, for any product of two or more integers, each integer is a factor of the total product. Also, the product of any combination of factors is a factor of the total product. This means that every one of the following products is a factor of 700. 0 0 6 0 0 0 8 0 0 60 0 0 60 0 0 0 0 and so on Proceeding in this manner, taking all combinations of the given factors two at a time, three at a time, and four at a time, we will find combinations. Not all of these combinations lead to different factors. For example, the list so far shows both 60 and 60. In fact, by using techniques involving prime factors (prime numbers are discussed in a future section), we can show that there are different factors of 700, including and 700 itself. Thus, by grouping the factors,,, 0, and in various ways and looking at the remaining factors, we can find the quotient if 700 is divided by any particular product of a group of the given factors. We have shown how this works in the case of 700 0 8. For 700, we can write 700 0 ( ) ( 0) 0, which shows that both and 0 are factors of 700 and 700 0. (Also, of course, 700 0.) Example 6 Divisibility of a Product Does divide the product 9 7? If so, how many times? Since 9 6 9, we factor as and rearrange the factors to find. 9 7 9 7 (9 ) ( 7) 70 Thus does divide the product, and it divides the product 70 times. Notice that we do not even need to know the value of the product, just the factors. 6 Chapter Fractions, Mixed Numbers, and Proportions
Example 7 Divisibility of a Product Does 6 divide the product 9 7? If so, how many times? 6 does not divide 9 7 because 6 and is not a factor of the product. Now work margin exercises 6 and 7. 6. Show that 9 divides the product 8 and find the quotient without actually dividing. 7. Explain why does not divide the product. 6. 8 ( ) 8 9 0 ( ) 7. 7 does not divide the product. Tests for Divisibility Section. 66
Exercises. For Using each the problem, tests for make divisibility, up any determine four -digit which numbers of the that numbers you can,, think, 6, of 9, and that 0 are (if divisible any) will by all divide of the exactly given into numbers. each of (There the following are many integers. possible See answers.) Examples through.. 8. 9. 8. 0, 9,,, 6, 0. 6. 7. 8. 70, 9,, 0 9. 7 0. 67. 7. 80,, 9, 9, 9. 60. 780. 6. 6,, 0,,, 6, 0 none none 7. 0 8. 70 9. 7890 0. 6790,, 6,, 6, 9,,, 6, 0,, 0. 777. 888.,000.,000,, 6,,, 6, 9, 0,, 0. 76 6. 8 7. 698 8. 80,, 9,, 6, 9,, 0 9. 9 0. 790.,.,87,,, 6, 9, 0,, 6 none.,77.,06. 7,8 6. 87,000 none,, 6,, 0 7. 6,000 8. 7,00 9. 7,69 0.,07,, 0,, 0 none. A test for divisibility by : If the number formed by the last two digits of a number is divisible by, then the number is divisible by. For example, is divisible by because is divisible by. (Mentally dividing gives. Using long division, we would find that 86. ) Use the test for divisibility by (just discussed) to determine which, if any, of the following numbers is divisible by. a. 8 yes b. 967 yes c. 86,6 no d. 9 no e. 78,6 yes. A digit is missing in each of the following numbers. Find a value for the missing digit so that the resulting number will be a. divisible by. (There may be several correct answers or no correct answer.) i. ii. iii. 7,88 iv. 7,7,, 8 0,, 6, 9 0,, 6, 9,, 7 b. divisible by 9. (There may be several correct answers or no correct answer.) i. ii. iii. 9, iv., 6 8 6 67 Chapter Fractions, Mixed Numbers, and Proportions
. A digit is missing in each of the following numbers. Find a value for the missing digit so that the resulting number will be a. divisible by 9. (There may be several correct answers or no correct answer.) i. ii., iii.,7 iv. 7, 7 7 7 8 b. divisible by 6. (There may be several correct answers or no correct answer.) i. ii., iii.,7 iv. 7,,, 7 none,, 8. Discuss the following statement: An integer may be divisible by and not divisible by 0. Give two examples to illustrate your point. Answers will vary. Any integer that ends in is divisible by and not divisible by 0. For example, and.. If an integer is divisible by both and 9, will it necessarily be divisible by 7? Explain your reasoning, and give two examples to support this reasoning. Answers will vary. Not necessarily. To be divisible by 7, an integer must have as a factor three times. Two counterexamples are 9 and 8. 6. If an integer is divisible by both and, will it necessarily be divisible by 8? Explain your reasoning and give two examples to support this reasoning. Answers will vary. Not necessarily. To be divisible by 8, an integer must have as a factor three times. Two counterexamples are and 0. Determine whether each of the given numbers divides (is a factor of) the given product. If so, tell how many times it divides the product. Find each product, and make a written statement concerning the divisibility of the product by the given number. See Examples 6 and 7. 7. 6; 7 The product is 6, and 6 divides the product times. 9. 0; 9 The product is 70, and 0 divides the product 7 times.. ; 7 0 The product is 00, and divides the product times. 8. ; 7 The product is 6, and divides the product 9 times. 0. ; 7 The product is 0, and since this product does not have as a factor twice, it is not divisible by.. ; 7 The product is, and since this product does not have as a factor, it is not divisible by.. ; 7 The product is 890, and divides the product times.. 0; 7 The product is 0, and 0 divides the product 6 times.. 6; The product is 60, and since this product does not have as a factor twice, it is not divisible by 6. 6. ; 7 0 The product is,00, and divides the product 90 times. Tests for Divisibility Section. 68
Writing & Thinking 7. a. If a whole number is divisible by both and, then it will be divisible by. Give two examples. b. However, a number might be divisible by and not by. Give two examples. c. Also, a number might be divisible by and not. Give two examples. 8. a. If a number is divisible by 9, then it will be divisible by. Explain in your own words why this statement is true. b. However, a number might be divisible by and not by 9. Give two examples that illustrate this statement. Collaborative Learning 9. In groups of three to four students, use a calculator to evaluate 0 0 and 0 0. Discuss what you think is the meaning of the notation on the display. (Note: The notation is a form of a notation called scientific notation and is discussed in detail in Chapter. Different calculators may use slightly different forms.) 7. a. 0, ; Answers will vary. b. 9, ; Answers will vary. c. 0, ; Answers will vary. 8. a. Because is a factor of 9, a number that is divisible by 9 will also be divisible by ; Answers will vary. b., 9; Answers will vary. 9. Answers will vary. The question is designed to help the students become familiar with their calculators and to provide an early introduction to scientific notation. 69 Chapter Fractions, Mixed Numbers, and Proportions
. Prime Numbers Objective A Prime Numbers and Composite Numbers The positive integers are also called the counting numbers (or natural numbers). Every counting number, except the number, has at least two distinct factors. The following list illustrates this idea. Note that, in this list, every number has at least two factors, but 7,, and 9 have exactly two factors. Counting Numbers Factors 7, 7,,,, 6,, 9, 9 0,,,, 0, 0,, 86,,, 86 Objectives A B C Understand the concepts of prime and composite numbers. Use the Sieve of Eratosthenes to identify prime numbers. Be able to determine whether a number is prime or composite. Our work with fractions will be based on the use and understanding of counting numbers that have exactly two different factors. Such numbers (for example, 7,, and 9 in the list above) are called prime numbers. (Note: In later discussions involving negative integers and negative fractions, we will treat a negative integer as the product of and a counting number.) Prime Numbers A prime number is a counting number greater than that has exactly two different factors (or divisors), itself and. Composite Numbers A composite number is a counting number with more than two different factors (or divisors). Thus, in the previous list, 7,, and 9 are prime numbers and, 0,, and 86 are composite numbers. Teaching Note: As a point of historical and practical interest, Euclid proved that there are an infinite number of prime numbers. As a direct consequence, there is no largest prime number. notes is neither a prime nor a composite number., and is the only factor of. does not have exactly two different factors, and it does not have more than two different factors. Prime Numbers Section. 70
. Explain why 9,, and are prime. Each has exactly two factors, and itself. Example Prime Numbers Some prime numbers: : has exactly two different factors, and. : has exactly two different factors, and. : has exactly two different factors, and. 7: 7 has exactly two different factors, and 7. Now work margin exercise.. Explain why,, and are composite.,, 7, and are all factors of.,,, 6, 7,,, and are all factors of.,,,, 6, and are all factors of. Example Composite Numbers Some composite numbers: :,,, and are all factors of. 9:,,, and 9 are all factors of 9. 9:, 7, and 9 are all factors of 9. 0:,,, 0,, and 0 are all factors of 0. Now work margin exercise. Objective B The Sieve of Eratosthenes One method used to find prime numbers involves the concept of multiples. To find the multiples of a counting number, multiply each of the counting numbers by that number. The multiples of,,, and 7 are listed here. Counting Numbers:,,,,, 6, 7, 8, Multiples of :,, 6, 8, 0,,, 6, Multiples of :, 6, 9,,, 8,,, Multiples of :, 0,, 0,, 0,, 0, Multiples of 7: 7,,, 8,,, 9, 6, All of the multiples of a number have that number as a factor. Therefore, none of the multiples of a number, except possibly that number itself, can be prime. A process called the Sieve of Eratosthenes involves eliminating multiples to find prime numbers as the following steps describe. The description here illustrates finding the prime numbers less than 0. Step : List the counting numbers from to 0 as shown. is neither prime nor composite, so cross out. Then, circle the next listed number that is not crossed out, and cross out all of its listed multiples. In this case, circle and cross out all the multiples of. 7 Chapter Fractions, Mixed Numbers, and Proportions
6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 Step : Circle and cross out all the multiples of. (Some of these, such as 6 and, will already have been crossed out.) 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 Step : Circle and cross out the multiples of. Then circle 7 and cross out the multiples of 7. Continue in this manner and the prime numbers will be circled and composite numbers crossed out (up to 0). 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 6 7 8 9 0 This last table shows that the prime numbers less than 0 are,,, 7,,, 7, 9,, 9,, 7,,, and 7. Two Important Facts about Prime Numbers. Even Numbers: is the only even prime number.. Odd Numbers: All other prime numbers are odd numbers. But, not all odd numbers are prime. Teaching Note: Students should be encouraged to memorize the list of prime numbers less than 0. We will assume later that, when reducing fractions, the students readily recognize these primes. Objective C Determining Prime Numbers An important mathematical fact is that there is no known pattern or formula for determining prime numbers. In fact, only recently have computers been used to show that very large numbers previously thought to be prime are actually composite. The previous discussion has helped to develop an understanding of the nature of prime numbers and showed some prime and composite numbers in list form. However, writing a list every time to determine whether a number is prime or not would be quite time consuming and unnecessary. The following procedure of dividing by prime numbers can be used to determine whether or not relatively small numbers (say, less than 000) are prime. If a prime number smaller than the given number is found to be a factor (or divisor), then the given number is composite. (Note: You still should memorize the prime numbers less than 0 for convenience and ease with working with fractions.) Prime Numbers Section. 7
Teaching Note: You might want to explain in more detail why only prime numbers are used as divisors to determine whether or not a number is prime. For example, if a number is not divisible by, then there is no need to try division by or 6 because a number divisible by or 6 would have to be even and therefore divisible by. Similarly, if a number is not divisible by, then there is no need to test 6 or 9. To Determine whether a Number is Prime Divide the number by progressively larger prime numbers (,,, 7,, and so forth) until:. You find a remainder of 0 (meaning that the prime number is a factor and the given number is composite); or. You find a quotient smaller than the prime divisor (meaning that the given number has no smaller prime factors and is therefore prime itself). (See example.) notes Reasoning that if a composite number was a factor, then one of its prime factors would have been found to be a factor in an earlier division, we divide only by prime numbers that is, there is no need to divide by a composite number.. Is 999 a prime number? No, since the sum of its digits is divisible by 9, 999 is divisible by 9. The sum of the digits is also divisible by. Example Determining Prime Numbers Is 0 a prime number? Since the units digit is, the number 0 is divisible by (by the divisibility test in Section.) and is not prime. 0 is a composite number. Now work margin exercise. Example Determining Prime Numbers Is 7 prime? Mentally, note that the tests for divisibility by the prime numbers,, and fail. Divide by 7: 7) 7 7 7 Chapter Fractions, Mixed Numbers, and Proportions
Divide by : 7 ) 7 Divide by : 9 7 ) 7 7 0 7 is composite and not prime. In fact, 7 9 and and 9 are factors of 7.. Is 89 a prime number? No, 7 7 89. Now work margin exercise. Example Determining Prime Numbers Is 09 prime? Mentally, note that the tests for divisibility by the prime numbers,, and fail.. Determine whether is prime or composite. Explain each step. is composite. Tests for,, and all fail. Next, divide by 7,, and. The first divisor of greater than is. Divide by 7: 7) 09 7 9 Divide by : 9 The quotient 9 is less than the divisor. ) 09 99 0 There is no point in dividing by larger numbers such as or 7 because the quotient would only get smaller, and if any of these larger numbers were a divisor, the smaller quotient would have been found to be a divisor earlier in the procedure. Therefore, 09 is a prime number. Now work margin exercise. Prime Numbers Section. 7
6. Determine whether 9 is prime or composite. Explain each step. 9 is prime. Tests for,, and all fail. Trial division by 7,,, and 7 fail to yield a divisor, and when dividing by 7, the quotient is less than 7. Completion Example 6 Determining Prime Numbers Is 99 prime or composite? Tests for,, and all fail. Divide by 7: 7) 99 Divide by : ) 99 Divide by : ) 99 Divide by : )99 99 is a number. Now work margin exercise 6. 7. Find two factors of 70 whose sum is 9. and Example 7 Applications One interesting application of factors of counting numbers (very useful in beginning algebra) involves finding two factors whose sum is some specified number. For example, find two factors for 7 such that their product is 7 and their sum is 0. The factors of 7 are,,,,, and 7, and the pairs whose products are 7 are 7 7 7 7. Thus the numbers we are looking for are and because 7 and + 0. Now work margin exercise 7. Completion Example Answers 8 6. Divide by 7: 7) 99 9 6 Divide by : ) 99 69 6 99 is a prime number. 8 Divide by : ) 99 89 88 Divide by 7: 7) 99 7 9 7 7 Chapter Fractions, Mixed Numbers, and Proportions
Exercises. List the first five multiples for each of the following numbers... 6. 8. 0., 6, 9,, 6,, 8,, 0 8, 6,,, 0 0, 0, 0, 0, 0,,,, 6. 7. 0 8. 0 9. 0. 6, 0,, 60, 7 0, 0, 60, 80, 00 0, 60, 90, 0, 0, 8,, 6,, 8, 6, 0, 0 First, list all the prime numbers less than 0, and then decide whether each of the following numbers is prime or composite by dividing by prime numbers. If the number is composite, find at least two pairs of factors whose product is that number. See Examples through 6.... 7. 67 prime prime prime prime. 6. 6 7. 97 8. 9 composite;, ;, composite;, 6;, prime prime 9. 0 0. 0. 79. 7 composite;, 0; composite;, 0; prime composite;, 7;,,, 7. 9. 07. 6. 96 composite;, 9; composite;, 07; composite;, ; composite;, 96;, 9, 7 9, 9, Two numbers are given. Find two factors of the first number such that their product is the first number and their sum is the second number. For example, for the numbers, 8, you would find two factors of whose sum is 8. The factors are and, since and + 8. See Example 7. 7., 8., 7 9. 6, 0 0., and 8 and and 8 and. 0,. 0, 9. 6,. 6, and 0 and and 9 and., 0 6. 7, 7., 0 8. 6, 8 and 7 and 9 and and 9. 7, 7 0., 7., 6. 66, 7 and and and 6 and. 6,. 6, 8. 7, 8 6. 60, 7 and and and and Prime Numbers Section. 76
7. Find two integers whose product is 60 and whose sum is 6. 6 and 0 8. Find two integers whose product is 7 and whose sum is 0. and 9. Find two integers whose product is 0 and whose sum is 8. and 0 0. Find two integers whose product is 0 and whose sum is 7. and 0. Find two integers whose product is 0 and whose sum is. and 0 Writing & Thinking. Discuss the following two statements: a. All prime numbers are odd. b. All positive odd numbers are prime.. a. Explain why is not prime and not composite. b. Explain why 0 is not prime and not composite.. Numbers of the form -, where N is a prime number, are sometimes prime. These prime numbers are called Mersenne primes (after Marin Mersenne, 88 68). Show that for N,,, 7, and the numbers - are prime, but for N the number - is composite. (Historical Note: In 978, with the use of a computer, two students at California State University at Hayward proved that, 70 the number - is prime.). a. False. The number is prime and even. b. False. Some counterexamples are 9,, and.. a. Answers will vary. The number does not have two or more distinct factors. b. Answers will vary. Since 0 is not a counting number, it does not satisfy the definition of either prime or composite.. N :, is prime. N : 8 7, 7 is prime. N :, is prime. N 7: 7 8 7, 7 is prime. N : 89 89, 89 is prime. N : 08 07, 07 is composite. The factors of 07 are:,, 89, 07. 77 Chapter Fractions, Mixed Numbers, and Proportions
. Prime Factorization Objective A Finding a Prime Factorization In all our work with fractions (including multiplication, division, addition, and subtraction), we will need to be able to factor whole numbers so that all of the factors are prime numbers. This is called finding the prime factorization of a whole number. For example, to find the prime factorization of 6, we start with any two factors of 6. 6 9 7 Although 7 is prime, 9 is not prime. Factoring 9 gives Objectives A B Be able to find the prime factorization of a composite number. Be able to list all factors of a composite number. 6 7. Now all of the factors are prime. This last product, 7, is the prime factorization of 6. Note: Because multiplication is a commutative operation, the factors may be written in any order. Thus we might write 6 7. The prime factorization is the same even though the factors are in a different order. For consistency, we will generally write the factors in order, from smallest to largest, as in the first case ( 7). notes For the purposes of prime factorization, a negative integer will be treated as a product of and a positive integer. This means that only prime factorizations of positive integers need to be discussed at this time. Regardless of the method used, you will always arrive at the same factorization for any given composite number. That is, there is only one prime factorization for any composite number. This fact is called the fundamental theorem of arithmetic. The Fundamental Theorem of Arithmetic Every composite number has exactly one prime factorization. Teaching Note: The skill of finding prime factorizations is absolutely indispensable in our work with fractions. You may want to emphasize this fact several times and to be sure students understand the distinction between the terms prime factorization and factors. Prime Factorization Section. 78
To Find the Prime Factorization of a Composite Number. Factor the composite number into any two factors.. Factor each factor that is not prime.. Continue this process until all factors are prime. The prime factorization of a number is a factorization of that number using only prime factors. Many times, the beginning factors needed to start the process for finding a prime factorization can be found by using the tests for divisibility by,,, 6, 9, and 0 discussed in Section.. This was one purpose for developing these tests, and you should review them or write them down for easy reference. Example Prime Factorization Find the prime factorization of 90. 90 9 0 OR 90 0 Since the units digit is 0, we know that 0 is a factor. 9 and 0 can both be factored so that each factor is a prime number. This is the prime factorization. is prime, but 0 is not. 0 0 is not prime. All factors are prime. Note: The final prime factorization was the same in both factor trees even though the first pair of factors was different. Since multiplication is commutative, the order of the factors is not important. What is important is that all the factors are prime. Writing the factors in order, we can write 90 or, with exponents, 90. 79 Chapter Fractions, Mixed Numbers, and Proportions
Example Prime Factorization Find the prime factorizations of each number. a. 6 6 b. 7 is a factor because the units digit is. Since both and are prime, is the prime factorization. 7 8 9 9 is a factor because the sum of the digits is 9. using exponents c. 9 9 7 9 7 7 7 is a factor because the units digit is even. is a factor of 7 because the sum of the digits is divisible by. using exponents If we begin with the product 9 69, we see that the prime factorization is the same. 9 6 9 7 7 7 Prime Factorization Section. 80
Completion Example Prime Factorization Find the prime factorization of 60. 60 6 using exponents Find the prime factorization of the following numbers.. 7. Completion Example Prime Factorization Find the prime factorization of 08.. 60 08. 66. 7 7 Now work margin exercises through. using exponents. Objective B Factors of a Composite Number. 60. 66 7 7 Once a prime factorization of a number is known, all the factors (or divisors) of that number can be found. For a number to be a factor of a composite number, it must be either, the number itself, one of the prime factors, or the product of two or more of the prime factors. (See the discussion of divisibility of products in section. for a similar analysis.) Factors of Composite Numbers The only factors (or divisors) of a composite number are:. and the number itself,. each prime factor, and. products formed by all combinations of the prime factors (including repeated factors). Completion Example Answers. 60 6 0 using exponents. 08 77 7 7 using exponents 8 Chapter Fractions, Mixed Numbers, and Proportions
Example Factorization Find all the factors of 60. Since 60, the factors are a. and 60 ( and the number itself), b.,, and (each prime factor), c., 6, 0,,, 0, and 0. Teaching Note: From number theory: The number of factors of a counting number is the product of the numbers found by adding to each of the exponents in the prime factorization of the number. For example, 60 ( + )( + )( + ) ()()() So, 60 has factors (as shown in Example ). The twelve factors of 60 are, 60,,,,, 6, 0,,, 0, and 0. These are the only factors of 60. Completion Example 6 Factorization Find all the factors of. The prime factorization of is 7. a. Two factors are and. b. Three prime factors are,, and. Find all the factors of the following numbers.. 6 6..,, 7, 9,, 6 6.,,, 6, 9, 8, 7, c. Other factors are. The factors of are. Now work margin exercises and 6. Completion Example Answers 6. a. Two factors are and. b. Three prime factors are, 7, and. c. Other factors are 7 7 77. The factors of are,,, 7,,,, and 77. Prime Factorization Section. 8
Exercises. In your own words, define the following terms.. prime number The prime number is a whole number greater than that has exactly two different factors (or divisors), itself and.. composite number A composite number is a counting number with more than two different factors (or divisors). Find the prime factorization for each of the following numbers. Use the tests for divisibility for,,, 6, 9, and 0, whenever they help, to find beginning factors. See Examples through.. 0.. 6. 0 7. 0 8. 6 9. 70 0. 80 0 6 70 7 80.. 6. 6. 7 6 6. 99 6. 9 7. 7 8. 99 9 7 7 is prime. is prime. 9. 0 0.. 96. 89 0 96 7 89 7. 6. 00. 6 6. 9 6 9 00 6 9 7 7. 000 8. 0,000 9. 00,000 0.,000,000 000 0,000 00,000,000,000 6 6. 600. 700. 07. 600 700 7 07 is prime. is prime.. 09 6. 0 7. 6 8. 09 0 0 0 6 7 9. 67 0.. 6. 67 6. 69. 00. 676 6. 77 69 7 00 676 77 9 8 Chapter Fractions, Mixed Numbers, and Proportions
For each of the following problems: a. Find the prime factorization of each number. See Examples through. b. Find all the factors (or divisors) of each number. See Examples and 6. 7. 8. 9. 00 0. 700 7,,, 6, 7,,,,,,, 6, 8,, 00,,,,, 6, 0,,, 0,, 0, 0, 60, 7, 00, 0, 00. 66. 96. 78. 0 66,,, 6,,,, 66 96,,,, 6, 8,,6,,, 8, 96 78,,, 6,, 6, 9, 78. 0 6. 7 7. 90 8. 7 0,,,, 6, 0,,, 0, 0, 7, 0 7 7,, 7,,, 7 90,,,, 6, 9, 0,, 8, 0,, 90 0,,, 0,, 6, 6, 0 7,,,,, 7 9. 00 60. 7 6. 8 6. 0 00 7, 7,,, 77, 9,, 00 7,,,,, 6,, 7 8,,, 9,,, 9,, 6, 7, 9, 8 0 7,,,, 6, 7, 0,,,,,, 0,,,,, 66, 70, 77, 0, 0,, 6, 0,, 0, 8, 6, 770,, 0 Collaborative Learning 700 7,,,, 7, 0,, 0,, 8,, 0, 70, 00, 0, 7, 0, 700 In groups of three to four students, discuss the theorem in Exercise 6 and answer the related questions. Discuss your answers in class. 6. In higher level mathematics, number theorists have proven the following theorem: Write the prime factorization of a number in exponential form. Add to each exponent. The product of these sums is the number of factors of the original number. For example, 60. Adding to each exponent and forming the product gives ( + ) ( + ) ( + ) and there are twelve factors of 60. Use this theorem to find the number of factors of each of the following numbers. Find as many of these factors as you can. a. 00 b. 660 c. 0 d. 8, 6. a. There are factors of 00. These factors are:,,,, 0, 0,, 0, 00,, 0, and 00. c. There are 8 factors of 0. These factors are:,,,, 6, 9, 0,, 8,, 0,, 0, 7, 90, 0,, and 0. b. There are factors of 660. These factors are:,,,,, 6, 0,,,, 0,, 0,,,, 60, 66, 0,, 6, 0, 0, and 660. d. There are 7 factors of 8,. These factors are: ; ; 7; ; ; ; 9; ; 77; ; 7; ; 7; 8; 9; 60; 87; ; 9; 69; 0; ; 99;,7;,7; 9,6; and 8,. Prime Factorization Section. 8
Objectives A B C Be able to use prime factorizations to find the least common multiple of a set of counting numbers. Be able to use prime factorizations to find the least common multiple of a set of algebraic terms. Recognize the application of the LCM concept in a word problem.. Least Common Multiple (LCM) Objective A Finding the LCM of a Set of Counting (or Natural) Numbers The ideas discussed in this section are used throughout our development of fractions and mixed numbers. Study these ideas and the related techniques carefully because they will make your work with fractions much easier and more understandable. Recall from a previous section that the multiples of an integer are the products of that integer with the counting numbers. Our discussion here is based entirely on multiples of positive integers (or counting numbers). Thus the first multiple of any counting number is the number itself, and all other multiples are larger than that number. We are interested in finding common multiples, and more particularly, the least common multiple for a set of counting numbers. For example, consider the lists of multiples of 8 and shown here. Counting Numbers:,,,,, 6, 7, 8, 9, 0,, D Be able to use prime factorizations to find the greatest common divisor of a set of counting numbers. Multiples of 8: 8, 6,,, 0, 8, 6, 6, 7, 80, 88, Multiples of :,, 6, 8, 60, 7, 8, 96, 08, 0,,... The common multiples of 8 and are, 8, 7, 96, 0,. The least common multiple (LCM) is. Listing all the multiples, as we just did for 8 and, and then choosing the least common multiple (LCM) is not very efficient. The following technique involving prime factorizations is generally much easier to use. To Find the LCM of a Set of Counting Numbers Teaching Note: You might want to inform students now that this is the technique for finding the least common denominator when adding and subtracting fractions. This knowledge may serve as a motivating factor for some of them.. Find the prime factorization of each number.. Identify the prime factors that appear in any one of the prime factorizations.. Form the product of these primes using each prime the most number of times it appears in any one of the prime factorizations. Your skill with this method depends on your ability to find prime factorizations quickly and accurately. With practice and understanding, this method will prove efficient and effective. STAY WITH IT! 8 Chapter Fractions, Mixed Numbers, and Proportions
notes There are other methods for finding the LCM, maybe even easier to use at first. However, just finding the LCM is not our only purpose, and the method outlined here allows for a solid understanding of using the LCM when working with fractions. Example Finding the Least Common Multiple Find the least common multiple (LCM) of 8, 0, and 0. a. prime factorizations: 8 three s 0 one, one 0 one, one, one b. Prime factors that are present are,, and. c. The most of each prime factor in any one factorization: Three s (in 8) One (in 0) One (in 0 and in 0) LCM 0 0 is the LCM and therefore, the smallest number divisible by 8, 0, and 0. Example Finding the Least Common Multiple Find the LCM of 7, 0,, and. a. prime factorizations: 7 three s 0 one, one, one 7 one, one 7 7 one, one, one 7 b. Prime factors present are,,, and 7. Least Common Multiple (LCM) Section. 86
c. The most of each prime factor in any one factorization is shown here. one (in 0 and in ) three s (in 7) one (in 0 and in ) one 7 (in and in ) LCM 7 7 890 890 is the smallest number divisible by all four of the numbers 7, 0,, and. Find the LCM for the following set of counting numbers.. 0, 68., 8, 6,. 0, 0, 70. 80. 6. 80 Completion Example Finding the Least Common Multiple Find the LCM for,, and 6. a. prime factorizations: 6 b. The only prime factors are,, and. c. The most of each prime factor in any one factorization: (in ) (in 6) (in ) LCM (using exponents) Now work margin exercises through. Completion Example Answers. a. prime factorizations: 6 b. The only prime factors are,, and. c. The most of each prime factor in any one factorization: three s (in ) two s (in 6) one (in ) LCM 60 87 Chapter Fractions, Mixed Numbers, and Proportions
In Example, we found that 890 is the LCM for 7, 0,, and. This means that 890 is a multiple of each of these numbers, and each is a factor of 890. To find out how many times each number will divide into 890, we could divide by using long division. However, by looking at the prime factorization of 890 (which we know from Example ) and the prime factorization of each number, we find the quotients without actually dividing. We can group the factors as follows: 890 ( ) ( 7 )( ) ( 7 ) So, 7 70 0 6 ( 7 ) ( )( 7 ) ( ) 7 divides into 890 70 times, 0 divides into 890 6 times, divides into 890 times, and divides into 890 times. Completion Example Finding the Least Common Multiple a. Find the LCM for 8, 6, and 66, and b. Tell how many times each number divides into the LCM. s a. 8 6 LCM 66 96 Teaching Note: Understanding the relationship between each number in a set and the prime factorization of the LCM of these numbers will increase the ease and speed with which students can perform addition and subtraction with fractions. In these operations, each denominator (and corresponding numerator) is multiplied by the number of times that the denominator divides into the LCM of the denominators. You may want to reinforce this idea several times.. Find the LCM for 0,, and 70; then tell how many times each number divides into the LCM. LCM 0 0 0 7 0 0 70 b. 96 ( ) 8 96 ( ) 6 96 ( ) 66 Now work margin exercise. Completion Example Answers.a. 8 6 LCM 66 96 b. 96 ( ) ( ) 8 96 ( ) ( ) 6 96 ( ) ( ) 66 6 Least Common Multiple (LCM) Section. 88
Objective B Finding The LCM of a Set of Algebraic Terms Algebraic expressions that are numbers, powers of variables, or products of numbers and powers of variables are called terms. Note that a number written next to a variable or two variables written next to each other indicate multiplication. For example,, ab, a, 6xy, 8x y are all algebraic terms. In each case where multiplication is indicated, the number is called the numerical coefficient of the term. A term that consists of only a number, such as, is called a constant or a constant term. Another approach to finding the LCM, particularly useful when algebraic terms are involved, is to write each prime factorization in exponential form and proceed as follows. Step : Find the prime factorization of each term in the set and write it in exponential form, including variables. Step : Find the largest power of each prime factor present in all of the prime factorizations. Remember that, if no exponent is written, the exponent is understood to be. Step : The LCM is the product of these powers. Example Finding the LCM of Algebraic Terms Find the LCM of the terms 6a, a b, a, and 8b. Write each prime factorization in exponential form (including variables) and multiply the largest powers of each prime factor, including variables. 6a a ab a b LCM a b 6ab a a 8b b 89 Chapter Fractions, Mixed Numbers, and Proportions
Example 6 Finding the LCM of Algebraic Terms Find the LCM of the terms 6x, xy, 0x y. Write each prime factorization in exponential form (including variables) and multiply the largest powers of each prime factor, including variables. 6x x xy x y 0x y x y LCM x y 00x y Completion Example 7 Finding the LCM of Algebraic Terms Find the LCM of the terms 8xy, 0x, and 0y. Find the LCM for each set of algebraic terms.. xy, 0y, x 6. ab, ab, 0a 8xy 0x 0y LCM 7. a, 6ab, b. 0xy 6. 60a b Now work margin exercises through 7. 7. 0ab Objective C An Application Many events occur at regular intervals of time. Weather satellites may orbit the earth once every 0 hours or once every hours. Delivery trucks arrive once a day or once a week at department stores. Traffic lights change once every minutes or once every minutes. The periodic frequency with which such events occur can be explained in terms of the least common multiple, as illustrated in Example 8. Completion Example Answers 7. 8xy x y 0x x 0y y LCM x 0x y y Least Common Multiple (LCM) Section. 90
8. A walker and two joggers begin using the same track at the same time. Their lap times are 6,, and minutes, respectively. a. In how many minutes will they be together at the starting place? Example 8 Weather Satellites Suppose three weather satellites A, B, and C are orbiting the earth at different times. Satellite A takes hours, B takes 8 hours, and C takes hours. If they are directly above each other now, as shown in part a. of the figure below, in how many hours will they again be directly above each other in the position shown in part a.? How many orbits will each satellite have made in that time? b. How many laps will each person have completed at this time? a. 0 minutes b., 0, and 6 A B C C B A B C A a. Beginning Positions b. Positions after 6 hours c. Positions after hours Study the diagrams shown above. When the three satellites are again in the initial position shown, each will have made some number of complete orbits. Since A takes hours to make one complete orbit, the solution must be a multiple of. Similarly, the solution must be a multiple of 8 and a multiple of to allow for complete orbits of satellites B and C. The solution is the LCM of, 8, and. 8 LCM 7 Thus the satellites will align again at the position shown in part a. in 7 hours (or days). Note that: Satellite A will have made orbits: 7 Satellite B will have made orbits: 8 7 Satellite C will have made 6 orbits: 6 7. Now work margin exercise 8. Objective D Greatest Common Divisor (GCD) Consider the two numbers and 8. Is there a number (or numbers) that will divide into both and 8? To help answer this question, the divisors for and 8 are listed on the next page. 9 Chapter Fractions, Mixed Numbers, and Proportions
Set of divisors for : {,,,, 6, } Set of divisors for 8: {,,, 6, 9, 8} The common divisors for and 8 are,,, and 6. The greatest common divisor (GCD) for and 8 is 6. That is, of all the common divisors of and 8, 6 is the largest divisor. Example 9 Finding All Divisors and the Greatest Common Divisor List the divisors of each number in the set {6,, 8} and find the greatest common divisor (GCD). Set of divisors for 6: {,,,, 6, 9,, 8, 6} Set of divisors for : {,,,, 6, 8,, } Set of divisors for 8: {,,,, 6, 8,, 6,, 8} The common divisors are,,,, 6, and. GCD. Now work margin exercise 9. 9. List the divisors of each number in the set {, 60, 90} and find the greatest common divisor (GCD). Set of divisors for : {,,, 9,, } Set of divisors for 60: {,,,,, 6, 0,,, 0, 0, 60} Set of divisors for 90: {,,,, 6, 9, 0,, 8, 0,, 90} GCD The Greatest Common Divisor The greatest common divisor (GCD)* of a set of natural numbers is the largest natural number that will divide into all the numbers in the set. As Example 9 illustrates, listing the divisors of each number before finding the GCD can be tedious and difficult. The use of prime factorizations leads to a simple technique for finding the GCD. Technique for Finding the GCD of a Set of Counting Numbers. Find the prime factorization of each number.. Find the prime factors common to all factorizations.. Form the product of these primes, using each prime the number of times it is common to all factorizations.. This product is the GCD. If there are no primes common to all factorizations, the GCD is. *The greatest common divisor is, of course, the greatest common factor, and the GCD could be called the greatest common factor, and be abbreviated GCF. Least Common Multiple (LCM) Section. 9
Example 0 Finding the GCD Find the GCD for {6,, 8}. 6 8 GCD In all the prime factorizations, the factor appears twice and the factor appears once. The GCD is. Example Finding the GCD Find the GCD for {60, 7, 0}. 60 6 0 9 7 GCD 0 6 Each of the factors and appears only once in all the prime factorizations. The GCD is. Example Finding the GCD Find the GCD for {68, 0, 0}. 68 8 7 0 0 6 7 7 0 6 6 7 GCD 7 8 In all prime factorizations, appears twice, once, and 7 once. The GCD is 8. If the GCD of two numbers is (that is, they have no common prime factors), then the two numbers are said to be relatively prime. The numbers themselves may be prime or they may be composite. 9 Chapter Fractions, Mixed Numbers, and Proportions
Example Relatively Prime Numbers Find the GCD for {, 8}. 8 GCD 8 and are relatively prime. Find the GCD for the following sets of numbers. Example Relatively Prime Numbers Find the GCD for {0, }. 0 7 GCD 0 and are relatively prime. Now work margin exercises 0 through. 0. {6,, 0}. {9,, 7}. {60, 90, 0}. {6, 9}. {, } 0. GCD 6. GCD 9. GCD 0. GCD. GCD Least Common Multiple (LCM) Section. 9
Exercises.. List the first twelve multiples of each number. a. b. 6 c., 0,, 0,, 0,, 6,, 8,, 0, 6,,, 0,, 60, 7, 90, 0, 0,, 0,, 60 8,, 60, 66, 7 0,, 0, 6, 80. From the lists you made in Exercise, find the least common multiple for the following pairs of numbers. a. and 6 0 b. and c. 6 and 0 Find the LCM of each of the following sets of counting numbers. See Examples through..,, 7 0., 7,. 6, 0 0 6. 9, 6 7.,, 66 8.,, 9 9.,, 0 0. 0,, 0 60. 0, 7 0. 0, 70 0. 0, 90 80. 0, 80 00. 8, 98 96 6. 0,, 0 7. 6,, 60 8.,, 98 9.,, 8 0.,, 90 90., 8, 6 6. 0, 0, 00 00. 0, 60, 0 0., 0, 7 0., 69, 96,9,6 6., 6, 8,00 7. 8, 6, 6 7,8,696 In Exercises 8 7, a. find the LCM of each set of numbers, and b. state the number of times each number divides into the LCM. See Example. 8. 0,, 9. 6,, 0 0. 0, 8, 90 LCM 0 LCM 0 LCM 90 0 0 0 6 0 6 0 0 90 0 9 8 90., 8, 7. 0, 8,. 99,, LCM 08 LCM 60 LCM,7 08 9 8 6 7 60 0 6 8 8,7 99 7 99.,,, 0. 0, 6, 60, 96 LCM 670 LCM 780 670 0 0 0 7 780 0 96 6 0 60 9 96 0 9 Chapter Fractions, Mixed Numbers, and Proportions
6., 9, 6, 6 7., 6, 98, 99 LCM 0 LCM 9,00 0 6 9 90 6 70 6 9,00 6 6 98 980 99 960 Find the LCM of each of the following sets of algebraic terms. See Examples through 7. 8. xy, 0xyz 00xyz 9. 0xyz, 7xy 600xy z 0. 0a b, 0ab 00a b. 6abc, 8a b a bc. 0x, x, 0xy 60x y. a, 0ab, b 0a b. 0xyz, xy, x z 700x y z. ab, 8abc, bc 8ab c 6. 6mn, m, 0mnp 0m np 7. 0m n, 60mnp, 90np 80m np 8. x y, 9xy, xy z 6x y z 9. 7xy z, 6xyz, x yz 08x y z 0. xyz, x, 0y 0x y z. ab, 66b, 76ab. x, x, 0x, 0x 600x. 0y, 0y, 0y, 0y 0y. 6a, 8a, 7a, 70a. c, 9c, 8 0c 6. 99xy, x, 6,7x y 7. 8abc, 7ax, 0ax y, a bc 90a bc x y 8. axy, x y, a by, 6a x,0a bx y Solve each of the following word problems. 9. Security: Three security guards meet at the front gate for coffee before they walk around inspecting buildings at a manufacturing plant. The guards take, 0, and 0 minutes, respectively, for the inspection trip. a. If they start at the same time, in how many minutes will they meet again at the front gate for coffee? 60 minutes b. How many trips will each guard have made?,, and trips, respectively 60. Aeronautics: Two astronauts miss connections at their first rendezvous in space. a. If one astronaut circles the earth every hours and the other every 8 hours, in how many hours will they rendezvous again at the same place? 90 hours b. How many orbits will each astronaut have made before the second rendezvous? 6 and orbits, respectively 6. Trucking: Three truck drivers have dinner together whenever all three are at the routing station at the same time. The first driver s route takes 6 days, the second driver s route takes 8 days, and the third driver s route takes days. a. How frequently do the three drivers have dinner together? every days b. How frequently do the first two drivers meet? every days 6. Lawn Care: Three neighbors mow their lawns at different intervals during the summer months. The first one mows every days, the second every 7 days, and the third every 0 days. a. How frequently do they mow their lawns on the same day? every 70 days b. How many times does each neighbor mow in between the times when they all mow together?, 9, and 6 times, respectively Least Common Multiple (LCM) Section. 96