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.