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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.

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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.

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