Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic
Any integer greater than 1 can be written as a unique product of prime numbers (ignoring the order of the prime numbers).
What does it mean?
"Any integer greater than 1" means the numbers 2, 3, 4, 5, 6, ... etc.
"Product of prime numbers" means that you multiply prime numbers together.
A prime number is a number that cannot be evenly divided by any other number (except 1 or itself).
The first few are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... (and more)
How about an example:
Example: 22 can be made by multiplying the prime numbers 2 and 11 together.
22 = 2 × 11
No other prime numbers would work.
So there are really two parts to this theorem:
1. You can make the number by multiplying prime numbers together.
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It is like the Prime Numbers are the basic building blocks of all numbers. |
Take any integer greater than 1, say 42. The theorem says we can make it by multiplying only prime numbers! Let's see:
Example: 42 can be made by multiplying the prime numbers 2, 3 and 7 together.
42 = 2 × 3 × 7
Well ... it works for 42. Try some other examples yourself!
And the other part is:
2. 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:
42 = 2 × 3 × 7
No other prime numbers would work!
You could try 2 × 3 × 5, or 5 × 11, and none of them will work:
Only 2, 3 and 7 make 42
Ignore the Order
At the top I said "ignoring the order of the prime numbers". By that I mean:
- 2 × 3 × 7 = 42 is the same as
- 7 × 3 × 2 = 42 or
- 2 × 7 × 3 = 42,
- etc.
So don't just rearrange the numbers and say "it isn't unique", OK?
Repeated Numbers
You 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 you would probably write it like this:
12 = 22 × 3
It is still a unique combination (2, 2 and 3)
Summary
The Fundamental Theorem of Arithmetic is like a "guarantee" that
any integer greater than 1 can be made by multiplying prime numbers
and
There is only way to do that in each case
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