We can take Cartesian product of given two dfas M1 & M2, accepting the languages L1 and L2 respectively where:
L1={w over Σ={a,b} accepting all strings which have number of a's divisible by 6}, say has 'm' states
L2={w over Σ={a,b} accepting all strings which have number of b's divisible by 8}, say has 'n' states
Step 1: Take Cartesian product and create a dfa M with m*n # of states. (total # of states)
Step 2: The difference of AND/OR comes on basis of selection of final states in M..
If OR(means UNION) so in M choose all states as final states wherever final states of EITHER M1 or M2 occurs..
If AND(means INTERSECTION) so in M choose all states as final states wherever final states of BOTH M1 and M2 occurs..
So by question, M1 has 6 states, M2 has 8 states. Final dfa has 6*8=48 states.(only total # of states is asked here and it will be minimum).
Please correct me if I am wrong.