smc

Every factor of $10^n$ will be of the form $2^a\times 5^b,a\le n,b\le n$. Not all of these base ten logarithms will be rational, but we can add them together in a certain way to make it rational. Recall the logarithmic property $\log (a\times b)=\log (a)+\log (b)$. For any factor $2^a\times 5^b$, there will be another factor $2^{n-a}\times 5^{n-b}$. Note this is not true if $10^n$ is a perfect square. When these are multiplied, they equal $2^{a+n-a}\times 5^{b+n-b}=10^n$. $\log 10^n=n$ so the number of factors divided by 2 times n equals the sum of all the factors, 792. There are $n+1$ choices for the exponent of 5 in each factor, and for each of those choices, there are $n+1$ factors (each corresponding to a different exponent of 2), yielding $(n+1{)}^2$ total factors. $\frac{(n+1{)}^2}{2}\ast n=792$. We then plug in answer choices and arrive at the answer $\boxed{11}$