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Finding factors of a number Use the beans to find factors of 24 Count out 24 beans We know that products can be illustrated using a rectangular model Make a rectangle using the beans What are the numbers you multiply to get 24? Can you arrange the beans into a different rectangle? What product does this represent?

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How many different rectangles can you make? Count out 11 beans. How many rectangles can you make with 11 beans?

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Sieve of Eratosthenes Eratosthenes was born in Cyrene which is now in Libya in North Africa in 276 BC. He died in 194 BC. Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. He was also fascinated with number theory, and he developed the idea of a sieve to illustrate prime numbers.

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The Sieve of Eratosthenes Prime Number Divisible only by 1 and itself Finding prime numbers using the sieve

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Sieve of Eratosthenes You will need many different colors. Use one color for each factor. Circle the number “1”. 1 is neither prime nor composite, as we have seen earlier. Now, circle 2. Every multiple of 2 is a composite number, so put a dot of that color next to all of the multiples of 2. Use a new color. Now, circle 3. Every multiple of 3 is a composite number, so put a dot of this new color next to all multiples of 3.

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Sieve of Eratosthenes Now, 4 has a dot next to it--it is not prime. Skip it and move on. Use a new color. Circle 5, and then put a dot of this new color next to all multiples of 5. Now, 6 has a dot next to it--it is not prime. Skip it and move on. Continue until you know that only prime numbers are left. When can you stop? How do you know?

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Sieve of Eratosthenes Questions to answer: When you circled 11, were there any multiples of 11 that did not already have dots next to them? Can you explain to a child why this was true? What does this have to do with factors and multiples? What are the prime numbers that are between 1 and 100? Is 1 a prime number?

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Sieve of Eratosthenes 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100

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Sieve of Erathosthenes

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Names for these numbers 11 is an example of a 24 is an example of a

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Factors of 24 List How should they be ordered? How do you know you have them all?

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Factors of 24--How do we know when we have them all? 1 12 2 24 3 8 4 6

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Exploration 4.2 First, fill in the table on page 85, using the information on the sieve. It will help if you write them in pairs. For example, for 18: 1, 18; 2, 9; 3, 6. The order does not matter. Next, fill in the table on page 87. Use the table on page 85 to help.

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Find all of the factors of 30

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Now find all of the factors of 60

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Factorization Factorization is writing a number as a product of factors. 24 60

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Prime Factorization A factorization of the number in which all of the factors are prime numbers. 10 12

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Prime Factorization 24 25

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Prime Factorization Using a factor tree to do prime factorization. 60

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112

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Exploration 4.3 is due on Thursday #1,2,6,7,8 along with some exercises from the textbook. Please put the exploration on a separate paper than the textbook problems.

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