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### Prime and Composite Numbers, Sieve of Eratosthenes

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Date: 01/28/97 at 16:35:06
From: Richard J. Principe
Subject: Prime numbers and composite numbers

I need to come up with a list of all prime numbers and all composite
numbers up to 50.  Can you help me?

Thank you very much.
Valerie
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Date: 02/02/97 at 15:36:44
From: Doctor Reno
Subject: Re: Prime numbers and composite numbers

Hi there, Valerie!

Before we talk about prime numbers, I want to review some
multiplication vocabulary with you...

Let's look at the multiplication problem 5 x 6 = 30.

The 5 and 6 are called factors. They are the numbers we multiply
together.

The answer 30 is called the product. 30 is also called a multiple
of 5 because it is a product of 5 and another number, 6.
30 is also a multiple of 6 because it is the product of 6 and 5.

The number 5 has many other multiples, like 10, 15, 20, 25, 35,
40, etc. The number 6 also has many other multiples, like 12, 18,
24, 36, 42, etc.

Notice that multiples are the "times tables" for a number.
30 is a multiple of not only 5 and 6, but also of 2 (2x15),
3 (3x10), 10, and 15.

A prime number is a numbers greater than one whose factors are only
one and itself. In other words, 6 is not prime, because its factors
are 1,2,3, and 6 (1x6, 2x3). But 5 is prime, because the only way you
can get a product of 5 is by multiplying 1 and 5 (1x5).

Composite numbers are all the other positive numbers greater than one.
6 is composite.

The number one is not prime OR composite because it has only one
factor.

Mathematicians have been fascinated by prime numbers for thousands of
years. In fact, Eratosthenes (275-194 BC, Greece), devised a "sieve"
to discover prime numbers. A sieve is like a strainer that you drain
spaghetti through when it is done cooking. The water drains out,
leaving your spaghetti behind. Eratosthenes's sieve drains out
composite numbers and leaves prime numbers behind! You can use the
sieve to find the prime numbers up to 100 (so you'll be ready for your
teacher's *next* question!).

To do what Eratosthenes did, make a chart of the first one hundred
whole numbers (1-100):

The Sieve of Eratosthenes

1   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

Next, cross out 1, because it is not prime.

Then circle 2, because it is the smallest positive even prime. Now
cross out every multiple of 2..; in other words, cross out every 2nd
number.

Then circle 3, the next prime. Then cross out all of the multiples of
3; in other words, every third number. Some, like 6, may have already
been crossed out since they may be multiples of 2.

Then circle the next open number, 5. Now cross out all of the
multiples of 5, or every 5th number.

Continue doing this until all the numbers through 100 are either
circled or crossed out!

And now, Valerie, if you have remembered your multiplication tables,
you have just circled all the prime numbers less than 100!

Computer people have written computer programs that do this same sieve
of Eratosthenes. They use the sieve to test computers and to tell them
how much faster one computer runs than another computer. You don't
have to stop the sieve at 100 - you can go up as far as you want to
find all the prime numbers you want to find. But, as you found, there
is a lot of multiplication involved. That is why computers are now
used to find prime numbers.

This page has some very difficult stuff on it, but at the end there
are links to other primes and their properties.

Take care, and have fun finding your prime numbers.

-Doctor Reno,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Elementary Prime Numbers