Find factors of the square of a number n that do not divide n and are less than n

Question

Let $n=2^{23}3^{17}$. How many factors of $n^2$ are less than $n$ but do not divide $n$?

I am unable to solve the problem above.

My question is, how does one determine what factors are less than $n$, considering directly comparing the numbers is not an option when we're working with very large ones in the problem?

Is there a method to compare the size of numbers based on their prime factorization?

Thank you very much in advance.

Answer 1

$n^2$ has $(46+1)(34+1)$ factors.
Of those, $\frac{(46+1)(34+1)+1}{2}$ of them are at most $n$.
Of those, $(23+1)(17+1)$ of them divide $n$.

Hence, there are ... (fill in the blank) factors of $n^2$ that are smaller than $n$ and do not divide $n$.