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 exactly 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?
 |
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 will 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 will 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 is not right, 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 yourself?
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 primes
and
There is only one way to do that in each case