Question 58
For any positive integers $a$ and $b$ with $b \gt 1$, let $s_{b}(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that
$$
\sum_{i=1}^{\left\lfloor\log _{23} n\right\rfloor} s_{20}\left(\left\lfloor\frac{n}{23^{i}}\right\rfloor\right)=103 \text{ and } \sum_{i=1}^{\left\lfloor\log _{20} n\right\rfloor} s_{23}\left(\left\lfloor\frac{n}{20^{i}}\right\rfloor\right)=115
$$
Compute $s_{20}(n)-s_{23}(n)$.
$$
\sum_{i=1}^{\left\lfloor\log _{23} n\right\rfloor} s_{20}\left(\left\lfloor\frac{n}{23^{i}}\right\rfloor\right)=103 \text{ and } \sum_{i=1}^{\left\lfloor\log _{20} n\right\rfloor} s_{23}\left(\left\lfloor\frac{n}{20^{i}}\right\rfloor\right)=115
$$
Compute $s_{20}(n)-s_{23}(n)$.