Question 1
An integer is square-free if it is not divisible by $a^2$ for any integer $a\gt 1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of
$$\sum_{k\in S}\left\lfloor\sqrt{\frac{10^{10}}{k}}\right\rfloor,$$
where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For example, $\lfloor2.5\rfloor=2$.
$$\sum_{k\in S}\left\lfloor\sqrt{\frac{10^{10}}{k}}\right\rfloor,$$
where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For example, $\lfloor2.5\rfloor=2$.