2023 SSMO Team Round Problems/Problem 13
Problem
Let 
denote the product of all divisors of 
Let 
denote the set of all integers that are both a multiple of 
and a factor of 
Let
\[
-F(a) = \sqrt{\left|\log_{10}\left(\frac{D(10^{a})}{\prod_{\omega\in P(10^2,10^{a+2})}\omega}\right)\right|}\text{ and }G(n) = \sqrt[n-1]{\prod_{i=2}^{n}10^{F(i)}}.
\]
Suppose 
is 
. Find the value of 
.
