1. The Sieve of Eratosthenes “Finding Prime Numbers” By: Patt Hawkey, MS

2. Vocabulary Used with Primes o Composite o Divisibility o Divisible o Even o Even Number o Factor o Factorization o Factor pair o Factor tree o GCF o LCM o Multiple o Odd o Odd Number o Prime o Prime Factorization o Test

3. What is a Sieve? o A colander or “sifter” often used in cooking. o The Sieve of Eratosthenes “Sifts” the numbers put into it to remove the multiples or the composite numbers and leaves the prime numbers.

4. A Sieve In Action!

5. Who Was Eratosthenes of Cyrene o He was a mathematician o Lived in 275 to 195 B.C. o First to estimate accurately the diameter of the earth. o Developed an interesting way of locating prime numbers which we call the: o Sieve of Eratosthenes

6. Counting or Natural Numbers o These are numbers starting at 1 o They can be classified as either a Prime or a Composite number. o A Prime Number is a number greater than one that has exactly 2 factors -- o Itself and one o A Composite Number has more than 2 factors o NOTE: Mathematicians decided, many years ago, that the number 1 will NOT be considered neither prime nor composite

7. Prime Factorization o Every natural number that is composite has only one prime factorization. For example: Hawkey Stick Factor Tree Method 12 2 6 2 3 2 x 2 x 3 12 6 2 2 3 2 x 2 x 3

8. The Sieve of Eratosthenes Method o It uses a hundreds chart where we will cross out the multiples with different colored pencils. o By counting by multiples to eliminate the composite numbers we will be left with only the primes on the chart. o Remember to look for patterns as you find the multiples or the composite numbers.

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

10. For this activity you will need: o 100 chart o Colored Pencils or Crayons: o black o red o green o blue o yellow

11. Finding the Primes Step #1 o Cross out in black the number one (1) - because it isn’t prime. o Circle the number 2, the first prime and using red cross out all of the multiples of 2

12. Finding the Primes Step # 2 o Circle the number 3 then using green cross out all of the multiples of 3 o Circle the number 5 then using blue cross out all of the multiples of 5

13. Finding the Primes Step # 3 o Circle the number 7, then using yellow cross out all of the multiples of 7 o Circle all of the remaining numbers that are not crossed out. They are prime numbers.

14. Sieve of Eratosthenes 25 Primes < 100 1 2 3 4 5 6 7 8 910 11 12 13 141516 17 18 19 20 2122 23 24 25 262728 29 30 31 323334 35 36 37 38 39 40 41 42 43 444546 47 48 49 50 5152 53 54 55 565758 59 60 61 626364 65 66 67 686970 71 72 73 747576 77 787980 81828384858687888990 91 9293949596979899100

15. Questions 1. Why didn’t we cross out multiples of 4 and 6? Because they are multiples of 2 2. Did we need to cross out the multiples of 8, 9 or 10? NO, because they are multiples of 2 and 3 3. What is the next prime number after 7? 11

16. Questions 4. What number do we get when we square that number? 121 5. Why could we stop after we crossed out multiples of 7? Because the square of 7 is 49 and 11 squared is 121 more than 100

17. Assignment o Answer the 17 questions on primes: “Prime Findin’”

18. Bibliography o http://www.math.utah.edu/~pa/Eratosthenes.html History and animation of the Sieve of Eratosthenes http://www.math.utah.edu/~pa/Eratosthenes.html o http://www.math.utah.edu/classes/216/assignment-07.html http://www.math.utah.edu/classes/216/assignment-07.html o http://www.faust.fr.bw.schule.de/mhb/eratosiv.htm Interactive animation of primes < 400 http://www.faust.fr.bw.schule.de/mhb/eratosiv.htm o http://www.mathnstuff.com/papers/games/prime.htm Prime Factor game http://www.mathnstuff.com/papers/games/prime.htm o http://www.research.att.com http://www.research.att.com o http://mathworld.wolfram.com http://mathworld.wolfram.com o http://mathforum.org - Ask Dr. Math http://mathforum.org