Prime Numbers
Numbers Divisible Only by Themselves and 1
Prime Numbers are the unique numbers that are
evenly divisible by ONLY themselves and the number 1.
Another way to
determine which numbers are prime is to use the concept of factors. A factor is any number that divides evenly into another number. For
instance 3 divides evenly into 15 to give 5 and conversely, 5 divides
evenly into 15 to give 3. So by the definition of a factor, both 3 and 5
are factors of 15.
Another definition is any number that has only two
factors, itself and 1.
The first five Prime Numbers are 2, 3, 5, 7 and 11. The number 1 is NOT
considered Prime because it only has one factor since "itself"
is = 1.
So, to determine if any number is Prime we only need to determine if it
has a factor (a number that it can be evenly divided by) other than
itself and 1. We will show you in the section below an easy method you
can use to determine if any number is prime.
The Eratosthenes' Sieve
Back in 200 BC a Greek mathematician named
Eratosthenes (pronounced er-uh-tos-thuh-neez) developed a
simple step-by-step method for determining which numbers are prime. This method
is called "Eratosthenes' Sieve".
A sieve is a strainer that allows liquid and small particles to pass
through while catching the larger particles. In the case of
Eratosthenes' Sieve, the small particles that pass through are all the
non-prime numbers and the large particles that remain are the Prime
Numbers. Eratosthenes' Sieve can be reduced to 6 simple steps.
The Eratosthenes' Sieve - Step 1
The first step of the Eratosthenes' Sieve is to make
a table of the numbers 1 to 100 organized into 10 rows as shown in the
picture above.
The Eratosthenes' Sieve - Step 2
The second step of the Eratosthenes' Sieve is to cross
out the number 1 since by definition it is NOT Prime as we discussed
above. This is shown in the picture above by the red X
The Eratosthenes' Sieve - Step 3
The third step of the Eratosthenes' Sieve is to leave 2 and cross out all numbers divisible by 2 as
shown in the picture above by the red X's. You will notice that when
the numbers are arranged in rows of 10 as in the picture above this
action allows you to eliminate all numbers of columns 2, 4, 6, 8 and 10
(except for the number 2 of course).
The Eratosthenes' Sieve - Step 4
The fourth step of the Eratosthenes' Sieve is to leave 3 and cross out all numbers divisible by 3 as
shown in the picture above by the red X's.
The Eratosthenes' Sieve - Step 5
The fifth step of the Eratosthenes' Sieve is to leave 5 and cross out all numbers divisible by 5 as
shown in the picture above by the red X's. As you can see this allows
you to cross out the remaining numbers in column 5 (except 5 of
course).
The Eratosthenes' Sieve - Step 6
The sixth and last step of the Eratosthenes' Sieve is to leave 7 and cross out all numbers divisible by 7 as
shown in the picture above by the red X's. As you can see this allows
you to cross out the remaining three numbers 49, 77 and 91.
Eratosthenes' Sieve-First 25 Prime Numbers
When the six steps explained above are carried out,
all the numbers between 1 and 100 that are not prime, go through the sieve and
the 25 Prime Numbers between 1 and 100 are caught in the sieve and
retained. These first 25 Prime Numbers are shown in red in the picture
above.
The Eratosthenes' Sieve - The Six Steps
The 6 steps of the Eratosthenes' Sieve are summarized
in the picture above. These steps can be done on only the first 1, 2, or 3 rows of the
Hundreds Table. You can also extend the method to numbers greater than
100 by adding more rows of 10 numbers and by replacing "7" in Step 6
with "the next Prime Number".
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