Factorization of element in quadratic integer ring

Question

Let $K=\mathbb{Q}(\sqrt{d})$ for square-free integer $d$ and $\mathcal{O}_K$ be the integral closure of $\mathbb{Z}$ in $K$. Assume that $(a)=P_1\ldots P_n$ for prime ideals $P_i$ such that $N(P_i)\leqslant m$ for $m>0$. Prove that $a$ is a product of some elements $\pi_i\in \mathcal{O}_K$ with $|N(\pi_i)|\leqslant m^{h_K}$, where $h_K$ is a class number of $K$.