Let $\mathfrak{m}$ be a nonzero integral ideal of the dedekind domain $\mathfrak{O}$. Show that in every ideal class of $Cl_K$, there exist an integral ideal prime to $\mathfrak{m}$.

My effort : Actually this is a problem given in Algebraic Number Theory by Neukrich(1.3.8). Let $\mathfrak{a}P_K\in Cl_K=J_K/P_K$. where $\mathfrak{a}$ is an fractional ideal of $K$. Hence there exist $c\in \mathfrak{O}$ such that $c\mathfrak{a}\subset \mathfrak{O}$. But after this point how to find $\mathfrak{b} \subset \mathfrak{O}$ such that $\mathfrak{m}+\mathfrak{b}=\mathfrak{O}$. Any help/hint in this regards would be highly appreciated. Thanks in advance!


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