[ColinWright]

Problem is, we only ever mess about with small numbers. Take some absolutely humungous number, N, and find, somehow, that it's a times b. Suppose someone else asserts that it's c times d. These numbers are enormous, absolutely enormous, so the question is: if a, b, c, and d are all prime, is it really obvious that these two ways of writing N have to be the same?

Edit: As pointed out elsewhere (and in the sibling to this comment) there are systems where "prime" and "irreducible" are not the same thing. In the integers they are, and that's sort-of why the FTA is true, but in other places they aren't, which is why the FTA is not obvious.