Prime Numbers and Divisibility

2.1 Prime Numbers and Divisibility 2.1 OBJECTIVES 1. Find the factors of a number 2. Determine whether a number is prime, composite, or neither 3. Determine whether a number is divisible by 2, 3, or 5 Overcoming Math Anxiety Working Together How many of your classmates do you know? Whether you are by nature gregarious or shy, you have much to gain by getting to know your classmates. 1. It is important to have someone to call when you have missed class or if you are unclear on an assignment. 2. Working with another person is almost always beneficial to both people. If you don t understand something, it helps to have someone to ask about it. If you do understand something, nothing will cement that understanding more than explaining the idea to another person. 3. Sometimes we need to commiserate. If an assignment is particularly frustrating, it is reassuring to find that it is also frustrating for other students. 4. Have you ever thought you had the right answer, but it doesn t match the answer in the text? Frequently the answers are equivalent, but that s not always easy to see. A different perspective can help you see that. Occasionally there is an error in a textbook (here, we are talking about other textbooks). In such cases it is wonderfully reassuring to find that someone else has the same answer as you do. NOTE 2 and 5 can also be called divisors of 10. They divide 10 exactly. In Section 1.5 we said that because 2 5 10, we call 2 and 5 factors of 10. Definitions: Factor A factor of a whole number is another whole number that will divide exactly into that number. This means that the division will have a remainder of 0. Example 1 Finding Factors List all factors of 18. NOTE This is a complete list of the factors. There are no other whole numbers that divide 18 exactly. Note that the factors of 18, except for 18 itself, are smaller than 18. 3 6 18 Because 3 6 18, 3 and 6 are factors (or divisors) of 18. 2 9 18 2 and 9 are also factors of 18. 1 18 18 1 and 18 are factors of 18. 1, 2, 3, 6, 9, and 18 are all the factors of 18. CHECK YOURSELF 1 List all the factors of 24. 129

130 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS Listing factors leads us to an important classification of whole numbers. Any whole number larger than 1 will be either a prime or a composite number. Let s look at the following definitions. A whole number greater than 1 will always have itself and 1 as factors. Sometimes these will be the only factors. For instance, 1 and 3 are the only factors of 3. Definitions: Prime Number NOTE How large can a prime number be? There is no largest prime number. To date, the largest known prime is 2 6972593 1. This is a number with 2,098,960 digits, if you are curious. Of course, a computer had to be used to verify that a number of this size is prime. By the time you read this, someone may very well have found an even larger prime number. A prime number is any whole number that has exactly two factors, 1 and itself. As examples, 2, 3, 5, and 7 are prime numbers. Their only factors are 1 and themselves. To check whether a number is prime, one approach is simply to divide the smaller primes, 2, 3, 5, 7, and so on, into the given number. If no factors other than 1 and the given number are found, the number is prime. Here is the method known as the sieve of Eratosthenes for identifying prime numbers. 1. Write down a series of counting numbers, starting with the number 2. In the example below, we stop at 50. 2. Start at the number 2. Delete every second number after the 2. 3. Move to the number 3. Delete every third number after 3 (some numbers will be deleted twice). 4. Continue this process, deleting every fourth number after 4, every fifth number after 5, and so on. 5. When you have finished, the undeleted numbers are the prime numbers. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 The prime numbers less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Example 2 Identifying Prime Numbers Which of the following numbers are prime? 17 is a prime number. 29 is a prime number. 33 is not prime. Note: For two-digit numbers, if the number is not a prime, it will have one or more of the numbers 2, 3, 5, or 7 as factors. CHECK YOURSELF 2 Which of the following numbers are prime numbers? 2, 6, 9, 11, 15, 19, 23, 35, 41 1 and 17 are the only factors. 1 and 29 are the only factors. 1, 3, 11, and 33 are all factors of 33.

PRIME NUMBERS AND DIVISIBILITY SECTION 2.1 131 We can now define a second class of whole numbers. Definitions: Composite Number NOTE This definition tells us that a composite number does have factors other than 1 and itself. A composite number is any whole number greater than 1 that is not prime. Every composite number has more than two factors. Example 3 Identifying Composite Numbers Which of the following numbers are composite? 18 is a composite number. 1, 2, 3, 6, 9, and 18 are all factors of 18. 23 is not a composite number. 1 and 23 are the only factors. This means that 23 is a prime number. 25 is a composite number. 1, 5, and 25 are factors. 38 is a composite number. 1, 2, 19, and 38 are factors. CHECK YOURSELF 3 Which of the following numbers are composite numbers? 2, 6, 10, 13, 16, 17, 22, 27, 31, 35 By the definitions of prime and composite numbers: Definitions: Zero and One The whole numbers 0 and 1 are neither prime nor composite. This is simply a matter of the way in which prime and composite numbers are defined in mathematics. The numbers 0 and 1 are the only two whole numbers that can t be classified as one or the other. For our work in this and the following sections, it is very useful to be able to tell whether a given number is divisible by 2, 3, or 5. The tests that follow will give you some tools to check divisibility without actually having to divide. Tests for divisibility by other numbers are also available. However, we have limited this section to those tests involving 2, 3, and 5 because they are very easy to use and occur frequently in our work. NOTE This, of course, means that the number is even. Rules and Properties: Divisibility by 2 A whole number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. Example 4 Determining If a Number Is Divisible by Two Which of the following are divisible by 2? 2346 is divisible by 2. 13,254 is divisible by 2. 23,573 is not divisible by 2. 57,085 is not divisible by 2.

132 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS CHECK YOURSELF 4 Which of the following are divisible by 2? 274 3587 7548 13,593 Rules and Properties: Divisibility by 3 A whole number is divisible by 3 if the sum of its digits is divisible by 3. Example 5 Determining If a Number Is Divisible by Three Which of the following are divisible by 3? 345 is divisible by 3. The sum of the digits, 3 4 5, is 12, and 12 is divisible by 3. 1243 is not divisible by 3. The sum of the digits, 1 2 4 3, is 10, and 10 is not divisible by 3. 25,368 is divisible by 3. The sum of the digits, 2 5 3 6 8, is 24, and 24 is divisible by 3. Note that 25,368 is also divisible by 2. CHECK YOURSELF 5 (a) Is 372 divisible by 2? By 3? (b) Is 5493 divisible by 2? By 3? Let s look now at a third divisibility test. Rules and Properties: Divisibility by 5 A whole number is divisible by 5 if its last digit is 0 or 5. Example 6 Determining If a Number Is Divisible by Five 2435 is divisible by 5. Its last digit is 5. 23,123 is not divisible by 5. Its last digit is 3. 123,240 is divisible by 5. Its last digit is 0. Do you see that 123,240 is also divisible by 2 and 3? CHECK YOURSELF 6 (a) Is 12,585 divisible by 5? By 2? By 3? (b) Is 5890 divisible by 5? By 2? By 3? CHECK YOURSELF ANSWERS 1. 1, 2, 3, 4, 6, 8, 12, and 24 2. 2, 11, 19, 23, and 41 are prime numbers 3. 6, 10, 16, 22, 27, and 35 are composite numbers 4. 274 and 7548 5. (a) Yes in both cases; (b) only by 3 6. (a) By 5 and by 3; (b) by 5 and by 2

Name 2.1 Exercises Section Date List the factors of each of the following numbers. 1. 4 2. 6 ANSWERS 3. 10 4. 12 5. 15 6. 21 1. 2. 3. 4. 7. 24 8. 32 9. 64 10. 66 11. 11 12. 37 5. 6. 7. 8. 9. Use the following list of numbers for exercises 13 and 14. 0, 1, 15, 19, 23, 31, 49, 55, 59, 87, 91, 97, 103, 105 13. Which of the given numbers are prime? 14. Which of the given numbers are composite? 15. List all the prime numbers between 30 and 50. 10. 11. 12. 13. 14. 15. 16. 17. 16. List all the prime numbers between 55 and 75. 18. Use the following list of numbers for exercises 17 through 20. 45, 72, 158, 260, 378, 569, 570, 585, 3541, 4530, 8300 17. Which of the given numbers are divisible by 2? 18. Which of the given numbers are divisible by 3? 133

ANSWERS 19. 20. 21. 22. A whole number is divisible by 5 if its last digit is 0 or 5, and divisible by 10 if its last digit is 0. Use this information in exercises 19 and 20. 19. Which of the given numbers are divisible by 5? 20. Which of the given numbers are divisible by 10? 23. 21. Seating Arrangements. A school auditorium is to have 350 seats. The principal wants to arrange them in rows with the same number of seats in each row. Use divisibility tests to determine if it is possible to have rows of 10 seats each. Are 15 rows of seats possible? 24. 25. 22. Group Size. Dr. Mento has a class of 80 students. For a group project, she wants to divide the students into groups of 6, 8, or 10. Is this possible? Explain your answer. 23. Use the sieve of Eratosthenes to determine all the prime numbers less than 100. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 24. Why is the following not a valid divisibility test for 8? A number is divisible by 8 if it is divisible by 2 and 4. Support your answer with an example. Give a valid divisibility test for 8. 25. Prime numbers that differ by two are called twin primes. Examples are 3 and 5, 5 and 7, and so on. Find one pair of twin primes between 85 and 105. 134

ANSWERS 26. The following questions refer to twin primes (see exercise 25). (a) Search for, and make a list of several pairs of twin primes, in which the primes are greater than 3. (b) What do you notice about each number that lies between a pair of twin primes? (c) Write an explanation for your observation in part (b). 26. 27. 27. Obtain (or imagine that you have) a quantity of square tiles. Six tiles can be arranged in the shape of a rectangle in two different ways: 28. 29. (a) Record the dimensions of the rectangles shown above. (b) If you use seven tiles, how many different rectangles can you form? (c) If you use ten tiles, how many different rectangles can you form? (d) What kind of number (of tiles) permits only one arrangement into a rectangle? More than one arrangement? 28. The number 10 has 4 factors: 1, 2, 5, and 10. We can say that 10 has an even number of factors. Investigate several numbers to determine which numbers have an even number of factors and which numbers have an odd number of factors. 29. Suppose that a school has 1000 lockers and that they are all closed. A person passes through, opening every other locker, beginning with locker #2. Then another person passes through, changing every third locker (closing it if it is open, opening it if it is closed), starting with locker #3. Yet another person passes through, changing every fourth locker, beginning with locker #4. This process continues until 1000 people pass through. (a) At the end of this process, which locker numbers are closed? (b) Write an explanation for your answer to part (a). (Hint: It may help to attempt exercise 28 first.) 135

Answers 1. 1, 2, 4 3. 1, 2, 5, 10 5. 1, 3, 5, 15 7. 1, 2, 3, 4, 6, 8, 12, 24 9. 1, 2, 4, 8, 16, 32, 64 11. 1, 11 13. 19, 23, 31, 59, 97, 103 15. 31, 37, 41, 43, 47 17. 72, 158, 260, 378, 570, 4530, 8300 19. 45, 260, 570, 585, 4530, 8300 21. Yes; No 23. 25. 27. 29. 136