Given a DFA A, give a formal construction of a PDA with three states M such that L(A) = L(M)
From the NFA $N$ construction an equivalent context-free grammar $G$ using the standard construction for this, and then construct a PDA $M$ such that $L(M) = L(G)$. The standard construction will (given a suitable definition of the notion of pushdown automaton) give us an automaton with three states only.

The automaton will simulate the leftmost derivations of $G$, and its stack symbols include the set $V$ of nonterminals and the set $\Sigma$ of terminals as well as a bottom marker $\$$. Initially, $\$$ is placed on the stack together with $S$. The main loop of the automaton is that involving the state $q_2$: If a nonterminal $A$ appears as the topmost symbol on the stack and there is a production $A \rightarrow w$ in $G$, then $A$ is replaced by $w$ in reverse. If a terminal $a$ appears as the topmost symbol on the stack and the next input symbol to be read is also $a$, then the automaton proceeds. Finally, if all input characters have been read and the bottom marker is the only symbol on the stack, the computation succeeds.