Prime Numbers and Composite Numbers

A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1.

Example: 7 can only be divided evenly by 1 or 7, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a composite number).

Let me explain ...

Some whole numbers can be divided up evenly, and some can't!

Example:

6 can be divided evenly by 2, or by 3:

6 = 2 × 3

Like this:

6 divided into 2 or 6 divided into 3

divided into 2 groups

 

divided into 3 groups

 

But 7 cannot be divided up evenly:

7 is Prime

And we give them names:

  • When a number can be divided up evenly it is a Composite Number
  • When a number can not be divided up evenly it is a Prime Number

So 6 is Composite, but 7 is Prime.

 

And that explains it ... but there are some more details ...

 

Not Into Fractions

We are only dealing with whole numbers here! We are not going to cut things into halves or quarters.

Not Into Groups of 1

OK, we could have divided 7 into seven 1s (or one 7) like this:

7 is Prime
7 = 1 x 7

But we could do that for any whole number!

So we should also say we are not interested in dividing by 1, or by the number itself.

It is a Prime Number when it can't be divided evenly by any number
(except 1 or itself).

 

Example: is 7 a Prime Number or Composite Number?

7 is Prime

  • We cannot divide 7 evenly by 2 (you would get 2 lots of 3, with one left over)
  • We cannot divide 7 evenly by 3 (you would get 3 lots of 2, with one left over)
  • We cannot divide 7 evenly by 4, or 5, or 6.

We can only divide 7 into one group of 7 (or seven groups of 1):

7 is Prime
7 = 1 x 7

 

So 7 can only be divided evenly by 1 or itself:

So 7 is a Prime Number

And also:

It is a Composite Number when it can be divided evenly
by numbers other than 1 or itself.

Like this:

Example: is 6 a Prime Number or Composite Number?

6 can be divided evenly by 2, or by 3, as well as by 1 or 6:

6 = 1 × 6
6 = 2 × 3

So 6 is a Composite Number

Sometimes a number can be divided evenly many ways:

Example: 12 can be divided evenly by 1, 2, 3, 4, 6 and 12:

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

So 12 is a Composite Number

And note this:

Any whole number greater than 1 is either Prime or Composite

What About 1?

Years ago 1 was included as a Prime, but now it is not:

1 is neither Prime nor Composite.

Factors

We can also define a Prime Number using factors.


"Factors" are numbers we multiply
together to get another number.

And we have:

When the only two factors of a number are 1 and the number,
then it is a Prime Number

It means the same as our previous definition, just stated using factors.

And remember this is only about Whole Numbers (1, 2, 3, ... etc), not fractions or negative numbers. So don't say "I could multiply ½ times 6 to get 3" OK?

Examples:

3 = 1 × 3
(the only factors are 1 and 3)
Prime
   
6 = 1 × 6   or   6 = 2 × 3
(the factors are 1,2,3 and 6)
Composite

Examples From 1 to 14

Factors other than 1 or the number itself are highlighted:

Number
Can be Evenly
Divided By
Prime, or
Composite?
1
(1 is not considered prime or composite)
2
1, 2
Prime
3
1, 3
Prime
4
1, 2, 4
Composite
5
1, 5
Prime
6
1, 2, 3, 6
Composite
7
1, 7
Prime
8
1, 2, 4, 8
Composite
9
1, 3, 9
Composite
10
1, 2, 5, 10
Composite
11
1, 11
Prime
12
1, 2, 3, 4, 6, 12
Composite
13
1, 13
Prime
14
1, 2, 7, 14
Composite
...
...
...

So when there are more factors than 1 or the number itself, the number is Composite.

A question for you: is 15 Prime or Composite?

Why All the Fuss about Prime and Composite?

Because we can "break apart" Composite Numbers into Prime Number factors.

2 and 2 and 3

It is like the Prime Numbers are the basic building blocks of all numbers.

And the Composite Numbers are made up of Prime Numbers multiplied together.

Here we see it in action:

2 is Prime, 3 is Prime, 4 is Composite (=2×2), 5 is Prime, and so on...

Example: 12 is made by multiplying the prime numbers 2, 2 and 3 together.

12 = 2 × 2 × 3

The number 2 was repeated, which is OK.

In fact we can write it like this using the exponent of 2:

12 = 22 × 3

 

And that is why they are called "Composite" Numbers because composite means "something made by combining things"

This idea is so important it is called The Fundamental Theorem of Arithmetic.

There are many puzzles in mathematics that can be solved more easily when you "break up" the Composite Numbers into their Prime Number factors. A lot of internet security is based on mathematics using prime numbers.