Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. For any prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is it true that every residue class in $\mathcal{O}_K/\mathfrak{p}$ contains an integer?. I can able to prove that it is true if $\mathfrak{p}$ is unramified and has inertial degree 1, but not for general prime ideals. Kindly request your answers to this problem.

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. For any prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is it true that every residue class in $\mathcal{O}_K/\mathfrak{p}$ contains an integer? I can prove that it is true if $\mathfrak{p}$ is unramified and has inertial degree 1, but not for general prime ideals. I kindly request your answers to this problem.