One easier way to solve this question is to take product of two DFAs given.
Product of M and N will give us DFA with 4 states: (A,C), (A,D), (B,C) and (B,D). In this (A,C) will be the initial state of DFA and (B,D) will be the final state (since we are taking intersection).
Now draw transition of this dfa using the given two DFA. We will find that no transition leads us to (B,D) final state. Which shows that intersection is none. Hence we need only one state to represent it.