Largest $n$ ?

$\begin{aligned} & \frac {\left( 2^{10^5} \right) \left( 5^{2^{10}} \right)} {10^{5^2}} \\ =& \frac {\left( 2^{10^5} \right) \left( 5^{2^{10}} \right)} {\left(2^{5^2}\right) \left(5^{5^2}\right)} \\ =& \frac {2^{10^5}} {2^{5^2}} \cdot \frac {5^{2^{10}}} {5^{5^2}} \\ =& 2^{10^5 - 5^2} \cdot 5^{2^{10} - 5^2} \\ =& 2^{100000-25} \cdot 5^{1024-25} \\ =& 2^{99974} \cdot 5^{999} \end{aligned}$

Therefore, the greatest value of $n$ is $\boxed{n = 999}$ .

Let $N = \dfrac {2^{10^5}\times 5^{2^{10}}}{10^{5^2}} = \dfrac {2^{100000} \times 5^{1024}}{10^{25}} = \dfrac {2^{98976} \times 10^{1024}}{10^{25}} = 2^{98976} \times 10^{999}$ . Therefore the largest divisor of $10^n$ is $10^{999}$ , that is $n=\boxed{999}$ .