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Problem 1 Students explore the number 1. The number 1 is not a prime Problem 1 The Game number nor is it a composite number. It is a factor of every A sieve is an old tool that is used to separate small particles from larger particles and is number, and the multiplicative usually a box with a screen for a bottom that allows the smaller pieces to fall through. identity. The Sieve of Eratosthenes screens out all of the composite numbers and leaves only the prime numbers. Prime numbers are numbers greater than 1 with exactly two distinct factors, 1 and the number itself. Composite numbers are numbers that have more than Grouping two distinct factors. You and your partner will use the Sieve of Eratosthenes to determine all of the prime numbers up to 100. Have students complete The figure shows the first 100 numbers written in numerical order in an array. Questions 1 through 5 with a partner. Then share the 1 2 3 4 5 6 7 8 9 10 responses as a class. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Share Phase, 31 32 33 34 35 36 37 38 39 40 Questions 1 through 5 41 42 43 44 45 46 47 48 49 50 • Why does the sieve work? 51 52 53 54 55 56 57 58 59 60 • Why do you think 61 62 63 64 65 66 67 68 69 70 Eratosthenes called his tool 71 72 73 74 75 76 77 78 79 80 a sieve? 81 82 83 84 85 86 87 88 89 90 • Why can you eliminate all 91 92 93 94 95 96 97 98 99 100 even numbers except 2 as primes? • What is the role of multiples 1. Start by putting a square around the number 1 because it is not a prime or composite in making the sieve work? number. 2. Circle the number 2 and cross out all of the multiples of 2. • How could you check to see 3. Circle the next number after 2 that is not crossed out. Then cross out all the multiples that the sieve found all the of that number that are not already crossed out. prime numbers under 100? 4. Continue in this fashion until you come to the first number greater than 10 that is not Which operations would crossed out. All of the remaining numbers have “fallen through the sieve” and are prime numbers. © 2011 Carnegie Learning you use? 5. List all of the prime numbers up to 100. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

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Talk the Talk The multiplicative identify is defined as the number 1. Talk the Talk Students state the characteristics of prime and ● The number 1 is neither prime nor composite. composite numbers. ● The number 1 is a factor of every number.

The number 1 is called the multiplicative identity. The multiplicative identity is the number 1. When it is multiplied by a second number, the product is the second number. Grouping An example is 1  5 5 5.

Ask a student to read the 1. Explain why the number 1 is neither prime nor composite. information aloud. Discuss The number 1 is not prime because it does not have two distinct factors: 1 and the definition and complete itself. It is not composite because it does not have more than two factors. One has Questions 1 through 3 as only itself as a factor, so it is neither prime nor composite. a class.

Discuss Phase, 2. State the characteristics prime numbers share. All prime numbers have exactly two distinct factors: 1 and the number itself. Talk the Talk • Besides the number 1, are any other numbers considered neither prime nor composite? 3. State the characteristics composite numbers share. • Can any shortcuts be used All composite numbers have more than two distinct factors. to determine if a number is a

34 • Chapter 1 Factors, Multiples, Primes, and Composites