For every positive integer $n$ we define $s_{n}$ as the sum of the digits of $n$ . Determine all pairs $(a, b)$ of positive integers for which $s_{ an + b} -s_{n}$ assumes a finite number of $n$ values in positive integers.
My outline is that $s _{10n} = s _{n}$ is trivial, or $s _{bᵏⁿ} = s _{n} $ on a base $b$ .
All pairs must be of the form $(10^k, b)$, where $0\leq b < 10^k$.
If $a$ is not a power of $10$, let $M$ be a power of $10$ which is larger than both $a$ and $b$. Consider $X_l = \sum_{i=0}^l M^i$, then $s_{aX_l+b} - s_{X_l}$ is different for different $l$.
If $a$ if a power of $10$ and $b \geq a$, then take $Y_l = 10^l-1$. $s_{aY_l+b} - s_{Y_l}$ is different for different $l$.