Well, 1 is the only positive integer that divides all other positive integers. And a number other than 1 is prime if its divisors are 1 and itself. It's easy to write these properties as formulas.
Here is more difficult exercise (for me of course) from definiable sets:
Let be the partial ordering (in a signature with ) whose elements are the positive integers, with iff divides .
Show that the set and the set of primes are both -definable in .
I don't have any idea how can I solve it. Could You give me some advices?
Any help will be highly appreciated.