Example of Localization and Prime Ideals

Question

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer.

Could an example have something to do with a UFD or Noetherian ring?

Any suggestions of an example would be great! Thanks.

Answer 1

Let $S=p_1⋃...⋃p_{n-1}$ where $p_i$'s are prime. then the primes of localization are $0,p_1,...,p_{n-1}$

Answer 2

Let $S$ be the multiplicatively closed set generated by all but $p-1$ prime numbers.