What is the possible number of states in the DFA of intersection of two given FA?
The product construction shows that if $L_1$ is accepted by a DFA having $n_1$ states and $L_2$ is accepted by a DFA having $n_2$ states then $L_1 \cap L_2$ is accepted by a DFA having $n_1n_2$ states; browse back to the product construction to understand why. The resulting DFA need not be minimal even if the DFAs for $L_1$ and $L_2$ were (a trivial example is given by $L_1 = L_2$, assuming $n_1&gt;1$). In your case, this means that if you can find a DFA for $\{w : 6\mid \#_a(w)\}$ having six states and one for $\{w : 8\mid \#_b(w)\}$ having eight states, then you can find a DFA for the intersection having 48 states.

Is this DFA minimal? You can use the Myhill&ndash;Nerode criterion to show that indeed it is: the 48 words $\{ a^i b^j : 0 \leq i &lt; 6, \; 0 \leq j &lt; 8 \}$ are pairwise incomparable.