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The Fundamental Theorem of Arithmetic (circa 300 BC)
ανψ ιντεγερ γρεατερ τηαν
Any integer greater than 1
1 ξαν βε ωριττεν ας α
can be written as a product
προδυξτι οφ πριμε
of prime numbers.
νυμβερς.
But why does this
matter ?
Euclid
Greek Mathematician
Ug
Typical Englishman of this
Period
All Integers can be made from primes
15 = 3 x 5
52 = 2 x 2 x 13
1000 = 2 x 2 x 2 x 5 x 5 x 5
123,456,789 = 3 x 3 x 3803 x 3607
2
3
5
Primes are the building blocks for all integers
How do we Find Prime Factors?
Split 18 into a pair
Draw out the factor tree
of factors
18
2 is prime, so this
branch is done
2
Split 9 into a pair of
factors
9
3
3
18 = 2 x 3 x 3
>>
>>>
How do we Find Prime Factors?
18
2 is prime, so this
branch is done
2
Split 9 into a pair of
factors
9
3
3
18 = 2 x 3 x 3
>>
How do we Find Prime Factors?
18
3 is prime, so this
branch is done
3
Split 6 into a pair of
factors
6
2
3
18 = 3 x 2 x 3
But how do we prove it for all Numbers ?
By contradiction !
1. If the theorem is not true, there must be a first number, which is not a prime
and can’t be written as a product of primes. We’ll call this number - First
Number.
2. All the numbers below First Number must obey the theorem, since they are
less than First Number.
3. All of these numbers are either prime or can be written as a product of primes.
Lets call these numbers - Lesser Numbers.
4. But First Number can’t be prime so it must have factors. The factors must be
less than First Number , so must be one of the Lesser Numbers.
5. Since we know that these factors are Lesser Numbers, which are either primes
or can be written as a product of primes, then we must be able to write First
Number as a product of primes !
This is a contradiction proving that the original
assumption made in point 1. must be false!