Prime & Composite Numbers

Definition: A prime number is a whole number with exactly two integral divisors, 1 and itself.

The number 1 is not a prime, since it has only one divisor.

So the smallest prime numbers are:

2, 3, 5, 7, ...

The number 4 is not prime, since it has three divisors (1, 2, and 4), and 6 is not prime, since it has four divisors (1, 2, 3, and 6).

Definition: A composite number is a whole number with more than two integral divisors.

So all whole numbers (except 0 and 1) are either prime or composite.

Example:

43 is prime, since its only divisors are 1 and 43.

44 is composite, since it has 1, 2, 4, 11, 22, and 44 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 2 which is a multiple of 2 is not a prime, since it has at least three divisors: 1, 2, and itself. (This means 2 is the only even prime.)

Any number greater than 3 which is a multiple of 3 is not a prime, since it has 1, 3, and itself as divisors. (For example, 303 is not prime, since 303 ÷ 3 = 101.)

Any number which is a multiple of 4 is also a multiple of 2, so we can rule these out.

Any number greater than 5 which is a multiple of 5 is not a prime. (So the only prime number ending with a 0 or 5 is 5 itself.)

Any number which is a multiple of 6 is also a multiple of 2 and 3, 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 119 prime?

First test for divisibility by 2. 119 is odd, so it's not divisible by 2.

Next, test for divisibility by 3. Add the digits: 1 + 1 + 9 = 11. Since 11 is not a multiple of 3, neither is 119. (Remember, this trick only works to test divisibility by 3 and 9.)

Since 119 doesn't end in a 0 or a 5, it's not divisible by 5.

Next, test for divisibility by 7. You'll find that 119 ÷ 7 = 17.

So the answer is NO... 119 is not prime.

Example 2:

Is 127 prime?

First test for divisibility by 2. 127 is odd, so it's not divisible by 2.

Next, test for divisibility by 3. Add the digits: 1 + 2 + 7 = 10. Since 10 is not a multiple of 3, neither is 127.

Since 127 doesn't end in a 0 or a 5, it's not divisible by 5.

Next, test for divisibility by 7. You'll find that 7 doesn't go in evenly.

The next prime is 11. But 11 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 (13, 17, 19, 23, etc.). The reason is that if 13 went in evenly, then we would have 127 = 13 × n for some number n. But then n would have to be less than 13... and we already know that 127 is not divisible by any number smaller than 13.

So the answer is YES... 127 is prime.

For more advanced topics and a list of the first 400 primes, go to the Prime Page or the page on prime factorization.