THE SIEVE OF ERATOSTHENES: Prime and Composite Numbers

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THE SIEVE OF ERATOSTHENESPrime and Composite
Numbers

• by Jan Harrison
• http//ellerbruch.nmu.edu/classes/CS255W04/cs255st
  udents/jharriso/P12/P12.html

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Eratosthenes

• Eratosthenes was a Greek scholar who lived in
  the 3rd century B.C. During his life, he made
  many important contributions to the fields of
  mathematics, geography, philosophy, and astronomy.

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Among his contributions to mathematics (over
2,000 years ago), Eratosthenes is particularly
well known for

• Calculating a surprisingly accurate measurement
  of the Earths circumference.
• Devising the Sieve of Eratosthenes, a tool for
  finding prime numbers.

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Prime Numbers

• A number is regarded as prime only if it has
  exactly two factors 1 and itself.
• For example
• The number 2 has exactly two factors 1 and
  itself (i.e., 1 and 2) therefore, it is a prime
  number.

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Composite Numbers

• If a number is not prime, then it is composite.
• For example
• The number 12 has six factors 1, 2, 3, 4, 6,
  and itself (i.e., 1, 2, 3, 4, 6, 12) therefore,
  it is a composite number.

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The number 1

• is neither prime nor composite.
• Why do you think this is? Hint Look back at
  the defini-tion of a prime num-ber.

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Prime or Composite?

• Is the number 22 prime or composite? Why?
• What about the number 39?
• 7?
• 77?

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The answers

• 22 is a composite number, as it has more than two
  factors (1, 2, 11).
• 39 is a composite number, as it has more than two
  factors (1, 3, 13).
• 7 is a prime number, as it has only two factors
  (1, 7).
• 77 is a composite number, as it has more than two
  factors (1, 7, 11).

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2 is the only even prime number. Why?
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Building Blocks

• Have you noticed that the factors of composite
  numbers are always prime numbers? The
  Fundamental Theorem of Arithmetic states that
  every natural number greater than 1 has one
  unique way of being represented as a product of
  its prime factors. For this reason, the prime
  numbers are sometimes called the building
  blocks of the natural numbers.

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Finding Prime Numbers

• The Sieve of Eratosthenes is a tool that will
  help you find prime numbers.
• Note A worksheet is available at the web page
  for this lesson, if you want to try this as you
  read along.

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The Sieve of Eratosthenes generally looks
something like this (depending on the range of
numbers included)
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How to use the Sieve of Eratosthenes

1. Since the number 1 is neither prime nor
   composite, draw a box around 1.
2. Circle the first unmarked number, 2, which is the
   first prime number. Then count by 2, crossing
   out each multiple of 2 (i.e., cross out 4, 6, 8,
   etc.the rest of the even numbers).
3. Circle the next unmarked number, 3. Then count
   by 3, crossing out each multiple of 3 that has
   not already been crossed out (i.e., 9, 15, 21,
   etc.).
4. Continue in this manner, circling primes and
   crossing out composites, until you are done
   sieving.

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Sieved-out Prime Numbers

• After you are done sieving, any numbers that
  remain (have not been crossed out) are prime
  numbers. Do you see any patterns in the Sieve of
  Eratosthenes?

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Patterns

• One of the educational advantages of the Sieve
  of Eratosthenes is that it helps to develop your
  ability to see and extend patterns. Note the
  even numbers, for example. Do you see an actual
  pattern reflected in the table? What about
  multiples of 5?

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Other ways to mark the prime and composite
numbers

• Circling and crossing off numbers is just one
  way to mark the prime and composite numbers in a
  Sieve of Eratosthenes. Some people color-code
  the numbers (e.g., marking the multiples of 2 in
  red, 3 in green, etc.), as in the earlier
  illustration. This helps them to find and
  analyze patterns. Can you think of other ways
  that you might mark the primes and composites?

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Now that youve learned about the Sieve of
Eratosthenes, do you see an error in this one
shown to you earlier?
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Just for fun (after trying the worksheet), check
out one or more of the following sites and their
interactive activities relating to primes and/or
the Sieve of Eratosthenes

• Sieve of Eratosthenes
• http//matti.usu.edu/nlvm/nav/frames_asid_158_g_4_
  t_1.html?openinstructions
• http//www.win.tue.nl/ida/demo/c1s4ja.html
• http//ccins.camosun.bc.ca/jbritton/sieve/jberato
  sapplet.htm
• Prime Number List
• http//ccins.camosun.bc.ca/7Ejbritton/jbprimelist
  .htm
• Prime Factorization Machine
• http//ccins.camosun.bc.ca/7Ejbritton/jbprimefact
  or.htm

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Sites to explore for more information on primes
or Eratosthenes

• The Prime Pages (University of TennesseeMartin)
• http//www.utm.edu/research/primes
• Prime Numbers
• http//www.factmonster.com/ipka/A0876084.html
• Worlds Largest Known Prime Number
• http//www.factmonster.com/ipka/A0920820.html
• Eratosthenes of Cyrene (University of St.
  Andrews)
• http//www-history.mcs.st-and.ac.uk/history/Mathem
  aticians/Eratosthenes.html

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• THE END