Is there a proof of the Fundamental Theorem of Arithemetic that does not make use of the Integers or Rational Numbers (as opposed to using only the Natural Numbers)? And if so, what is it?
By the Fundamental Theorem of Arithemetic, I mean that any natural number that is greater than 1 is a product of irreducible natural numbers and that this product is unique up to the order of multiplication.
The part that poses difficulty is proving is that if an irreducible number divides a product of two numbers, then it divides at least one of these numbers, which is used to show the uniqueness. Euclid makes use of rational numbers, and the only other proof that I have seen uses the lemma that if two natural numbers $n$ and $m$ are coprime, then there exists integers $l$ and $k$ such that $ln + km = 1$, which involves Integers rather than just Natural Numbers.