Definition: A prime number is a whole number with exactly two integral divisors, and itself.
The number is not a prime, since it has only one divisor.
So the smallest prime numbers are:
The number is not prime, since it has three divisors ( , , and ), and is not prime, since it has four divisors ( , , , and ).
Definition: A composite number is a whole number with more than two integral divisors.
So all whole numbers (except and ) are either prime or composite.
Example:
is prime, since its only divisors are and .
is composite, since it has and as divisors.
How can you tell if a number is prime?
First of all, here are some ways to tell if a number is NOT prime:
Any number greater than which is a multiple of is not a prime, since it has at least three divisors: , , and itself. (This means is the only even prime.)
Any number greater than which is a multiple of is not a prime, since it has , and itself as divisors. (For example, is not prime, since .)
Any number which is a multiple of is also a multiple of , so we can rule these out.
Any number greater than which is a multiple of is not a prime. (So the only prime number ending with a or is itself.)
Any number which is a multiple of is also a multiple of and , so we can rule these out too.
You can continue like this... basically, you just have to test for divisibility by primes!
Example 1:
Is prime?
First test for divisibility by . is odd, so it's not divisible by .
Next, test for divisibility by . Add the digits: . Since is not a multiple of , neither is . (Remember, this trick only works to test divisibility by and .)
Since doesn't end in a or a , it's not divisible by .
Next, test for divisibility by . You'll find that .
So the answer is NO... is not prime.
Example 2:
Is prime?
First test for divisibility by . is odd, so it's not divisible by .
Next, test for divisibility by . Add the digits: . Since is not a multiple of , neither is .
Since doesn't end in a or a , it's not divisible by .
Next, test for divisibility by . You'll find that doesn't go in evenly.
The next prime is . But doesn't go in evenly, either.
You can stop now... it must be prime! You don't need to keep checking for divisibility by the next primes ( etc.). The reason is that if went in evenly, then we would have for some number . But then would have to be less than ... and we already know that is not divisible by any number smaller than .
So the answer is YES... is prime.
For more advanced topics and a list of the first primes, go to the Prime Page or the page on prime factorization .