Prime and Composite Numbers (2024)

Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,more...

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A Prime Number is:

a whole number above 1 that cannot be made by multiplying other whole numbers

Not 1

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

1 is not Prime and also not Composite.

Dividing Into Equal Groups

It is all about trying to divide the number into equal groups

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

And we give them names:

• When a number can be divided up exactly it is a Composite Number
• When a number cannot be divided up exactly it is a Prime Number

So 6 is Composite, but 7 is Prime

Like this:

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 = 1 x 7

But we could do that for any whole number!

So we are only interested in dividing by whole numbers other than the number itself.

Example: is 7 a Prime Number or Composite Number?

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

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

7 = 1 x 7

So 7 is a Prime Number

And also:

It is a Composite Number when it can be divided exactlyby a whole number other than itself.

Like this:

Sometimes a number can be divided exactly in many ways:

And note this:

Activity

Factors

We can also define a Prime Number using factors.

"Factors" are numbers we multiply
togetherto getanother 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:

Examples From 1 to 14

Factors other than 1 or the number itself are highlighted:

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.

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...

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 we "break up" the Composite Numbers into their Prime Number factors.

And a lot of internet security is based on mathematics using prime numbers in a subject called cryptography.

369, 1692, 1054, 1693, 2982, 2983, 2984, 3976, 2985, 3977