How to propose a method to construct an automaton?

Given $$0 \leq m < k$$ and $$p \geq 2$$ it's defined $$A_{k,m,p} = \{ \alpha \in \{0,1,\dots,p-1\}^* | \alpha \text{ is a p-ary representation of } x \backepsilon x \text{ mod } k = m \}$$

It is needed to give a general method to construct an automaton which accepts this language. I'm trutly lost about the procedure in this. Any hint would be helpful.

Some observations about this: The automaton is not specified (i.e. could be NFA or DFA) and the numbers are static.

Your automaton will have $$k$$ states, one for each possible value of $$m$$, i.e., one for each of the congruence $$(\text{mod } k)$$ classes. Exactly one of these states will be marked as the final state, because that will be the one that accepts ocurrences of $$\alpha$$ that are compatible with $$x ( \text{mod } k) =m$$.
The transition function will connect states representing congruence classes for $$m=a$$ and $$m=b$$ by an arrow labeled with a symbol denoting $$c$$, if and only if $$(a.p +c)\text{ mod } k = b$$.
The initial state will be the one corresponding to $$m = 0$$, assuming leading zeros are allowed.