Here is more difficult exercise (for me of course) from definiable sets:

Let $\displaystyle A$ be the partial ordering (in a signature with $\displaystyle \leq$) whose elements are the positive integers, with $\displaystyle m \leq ^A$ iff $\displaystyle m$ divides $\displaystyle n$.

Show that the set $\displaystyle \{1\}$ and the set of primes are both $\displaystyle \emptyset$-definable in $\displaystyle A$.

I don't have any idea how can I solve it. Could You give me some advices?

Any help will be highly appreciated.