Fundamental Theorem of Arithmetic

The Basic Idea

The Basic Idea is that any integer above 1 is either a Prime Number, or can be made by multiplying Prime Numbers together. Like this:

This continues on:

  • 10 is 2×5
  • 11 is Prime,
  • 12 is 2×2×3
  • 13 is Prime
  • 14 is 2×7
  • 15 is 3×5
  • 16 is 2×2×2×2
  • 17 is Prime
  • etc...

So they are either prime, or primes multiplied together

Read on for an explanation ...

The Fundamental Theorem of Arithmetic

Let us start with the definition:

Any integer greater than 1 is either a prime number, or can be written as a unique product of prime numbers (ignoring the order).

What does it mean?

Let's build up the ideas piece by piece:

"Any integer greater than 1" means the numbers 2, 3, 4, 5, 6, ... etc.

A Prime Number is a number that cannot be evenly divided by any other number (except 1 or itself).

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... (and more)

"...product of prime numbers" means that we multiply prime numbers together.

So, by multiplying prime numbers we can create any other whole number.

Example: 42

Can we make 42 by multiplying only prime numbers? Let's see:

2 × 3 × 7 = 42

Yes, 2, 3 and 7 are prime numbers, and when multiplied together they make 42.

Try some other examples for yourself. How about 30? Or 33?

2 and 2 and 3

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

"... unique product of prime numbers" means there is only one (unique!) set of prime numbers that will work

Example: we just showed that 42 is made by the prime numbers 2, 3 and 7:

2 × 3 × 7 = 42

No other prime numbers would work!

We could try 2 × 3 × 5, or 5 × 11, but none of them will work:

Only 2, 3 and 7 make 42

 

So there you have it!

Any of the numbers 2, 3, 4, 5, 6, ... etc are either prime numbers, or can be made by multiplying prime numbers together.

And there is only one (unique) set of prime numbers that works in each case.

More examples:

Example: 7

7 is already a prime number

Example: 22

22 can be made by multiplying the prime numbers 2 and 11 together.

2 × 11 = 22

No other combination of prime numbers would work.

Ignore the Order

Also, at the top I said "ignoring the order". By that I mean:

  • 2 × 11 = 22 is the same as
  • 11 × 2 = 22

So don't just rearrange the numbers and say "it isn't unique", OK?

Repeated Numbers

We may have to repeat a prime number!

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

12 = 2 × 2 × 3

That is OK. In fact we can write it like this:

12 = 22 × 3

It is still a unique combination (2, 2 and 3)

(Note: 4 × 3 does not work, as 4 is not a prime number)

The First Few

2
Is a Prime
3
Is a Prime
4
= 2×2 = 22
5
Is a Prime
6
= 2×3
7
Is a Prime
8
= 2×2×2 = 23
9
= 3×3 = 32
10
= 2×5
11
Is a Prime
12
= 2×2×3 = 22×3
13
Is a Prime
14
= 2×7
...
...

Why not continue this list to 100 as an exercise ...

Summary

The Fundamental Theorem of Arithmetic is like a "guarantee"
that any integer greater than 1
is either prime
or can be made by multiplying prime numbers

and

There is only way to do that in each case