Slide 2:
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
Slide 3:
How do we Find Prime Factors ? 18 2 9 3 3 Draw out the factor tree Split 18 into a pair of factors 2 2 is prime, so this branch is done Split 9 into a pair of factors 3 3 18 = 2 x 3 x 3 >> >>>
Slide 4:
How do we Find Prime Factors? 18 2 9 3 3 2 2 is prime, so this branch is done Split 9 into a pair of factors 3 3 18 = 2 x 3 x 3 >>
Slide 5:
How do we Find Prime Factors? 18 3 6 2 3 3 3 is prime, so this branch is done Split 6 into a pair of factors 2 3 18 = 3 x 2 x 3
Slide 6:
But how do we prove it for all Numbers ? By contradiction ! 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 . All the numbers below First Number must obey the theorem, since they are less than First Number. All of these numbers are either prime or can be written as a product of primes. Lets call these numbers - Lesser Numbers. 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. 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!