Fundamental theorem of arithmetic
In mathematics, and in particular number theory, the fundamental
theorem of arithmetic or unique factorization theorem is the statement
that every positive integer greater than 1 can be written as a product of
prime numbers in only one way. For instance, we can write
6936 = 23 x 3 x 172 or 1200 = 24 x 3 x 52
and there are no other possible factorizations of 6936 or 1200 into prime
numbers, if we ignore the ordering of the factors. To make the theorem
work even for the number 1, we can think of 1 as being the product of
zero prime numbers
Do you think that it is true that every (non-prime) number can indeed be
written as a product of primes? How could we come to know that this is
the case? Could we look at every number and check it? Could we show
that it must apply to every number, even those we shall never get to –
demonstrate that it is a necessary truth?
Suppose there were a positive integer which can not be written as a
product of primes.
P1
Then there must be a smallest such number: let's call it n. This
number n cannot be 1, because of our convention above.
P2
It cannot be a prime number either, since any prime number is a
product of a single prime, itself. So it must be a composite number.
P3
Thus n = ab where both a and b are positive integers smaller than
n.
P4
Since n was the smallest number for which the theorem fails, both
a and b can be written as products of primes. But then n = ab can
be written as a product of primes as well, a contradiction.