Given a real number \(x,\) let \(\lfloor x \rfloor\) denote the greatest integer less than or equal to \(x.\) For a certain integer \(k,\) there are exactly \(70\) positive integers \(n_{1}, n_{2}, \ldots, n_{70}\) such that \(k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor\) and \(k\) divides \(n_{i}\) for all \(i\) such that \(1 \leq i \leq 70.\) Find the maximum value of \(\frac{n_{i}}{k}\) for \(1\leq i \leq 70.\)

(第二十五届AIME2 2007 第7题)