ABSTRACT
This chapter aims to prove that the factorization into primes can be done in only one way. The Fundamental Theorem of Arithmetic (FTA) says that any positive integer greater than 1 is either prime or can be factored in exactly one way as a product of primes. Put more colloquially, this says that every integer greater than 1 is built up out of primes in exactly one way. This type of theorem is called an existence-uniqueness theorem because it makes two assertions. The first, the existence part, is that every integer can be written as a product of prime numbers. This will be relatively easy to show. The second, the uniqueness part, says that this can be done in only one way. This is harder to show and its proof relies on the Euclidean Algorithm. There is enough information in Euclid's Elements to deduce the FTA.
