I have thought a lot about this and the only thing that occurs to me is that $(\left \{2,5\right \}, +, \cdot, 0)$ is a submodel, where $+^\left \{2,5\right \}=+^(\mathbb{N} ,+, \cdot , 0) \mid_{\mathbb{N}^2}$ and the same for "$\cdot$", I'm thinking correctly, how many more are there, how can I find them? What you think?
Hint: You need $0$ in you submodel to satisfy (b). Then you need closure under addition and multiplication, so you need $4,7,10,9$ and other numbers in your submodel. If you just close $\{0,2,5\}$ under addition and multiplication you have most of the naturals. Which ones are missing? If you add in one of the missing ones, you need to have closure under addition and multiplication, so you don't have all the options you might think.