Question 7
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 $$
\begin{aligned} k &= \lfloor\sqrt[3]{n_1}\rfloor = \lfloor\sqrt[3]{n_2}\rfloor \\ &= \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor \end{aligned}
$$ and $k$ divides $n_i$ for all $i$ such that $1 \le i \le 70.$ Find the maximum value of $\frac{n_i}{k}$ for $1 \le i \le 70.$
\begin{aligned} k &= \lfloor\sqrt[3]{n_1}\rfloor = \lfloor\sqrt[3]{n_2}\rfloor \\ &= \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor \end{aligned}
$$ and $k$ divides $n_i$ for all $i$ such that $1 \le i \le 70.$ Find the maximum value of $\frac{n_i}{k}$ for $1 \le i \le 70.$