first of all we have to think this kind of problem in mathematical way..now [any number here it is no of a i.e. n(a)] mod 3 it will give result {0 or 1 or 2}..similarly [any number here it is no of b i.e. n(b) ]mod 2 ..will give result { 0 or 1}.now we have to think how should we can relate two different result.by relation yes,now we have to multiply both sets and it will give result {0,0} ,{0,1},{1,0},{1,1},{2,0},{2,1}.now we have to take only 6 states in our finite state machine for comparison between n(a)mod 3 and n(b) mod2.now draw 6 states dfa by counting a and b and take all states as final states where namod3>=nbmod2 .these states are{ 0,0 },{1,0},{ 1,1},{ 2,0},{ 2,1}. 
