I hope this isn't a stupid question...

It's well known that (in the presence of various other axioms), Euclid's Postulate 5 ('parallel axiom') is equivalent to the Pythagorean Theorem. That is, assume Pyth. Thm. is true without Postulate 5, and you get the 'parallel axiom' as a theorem.

My question: Are there well-known (or not-so-well-known) theorems/properties of the ring of integers which are *equivalent* to the Fundamental Theorem of Arithmetic in this way? That is, things which are not just consequences of it, but imply it.

I have had a lot of trouble finding anything about this on the Net, but of course the words involved are not exactly unique! Please be gentle if there is something obvious I'm missing - I've put "elementary" as a tag by way of anticipating there is a clear answer. At least the statement is elementary!

Edit: I like all three answers for different reasons, and have voted up accordingly. None really answers my question, but that's because, upon further review, I think it's not well posed. After all, FTA is not an axiom like Postulate 5 (though of course one needs various axioms to prove it).

So maybe the answer about $a|bc$ is closest to what I was looking for, though as it happens I like to prove this first as well. Probably the best question would be how much one can prove in number theory without using the FTA. But that would be a different question! Thanks.