Using standard notations, let $K$ be a number field and $S = \left\{p_{1}, ..., p_{n}\right\}$ a finite set of non-zero prime ideals of $K$. Let $a$ be a non-zero fractional ideal of $K$. Prove that there exists $\alpha \in K^{\times}$ such that $b := \alpha a$ has prime factorization $b = \prod_{p \in S}{p^{n_{p}(b)}},$ with $n_{p}(b)=0$ for all $p_i$'s in $S$.
I know this should follow from the unique factorization directly somehow, but I'm traving trouble picking the $\alpha$...