We need a DFA for:
$L = \{w | \neg\,(n_a(w)\, mod \,3 =0) \,\text{or} \,\neg\,(n_b(w) <= 4)\}$
$L = \{w | \neg\,[(n_a(w)\, mod \,3 =0) \,\text{and} \,\,(n_b(w) >= 5)]\}$
$\bar{L} = \{w | (n_a(w)\, mod \,3 =0) \,\text{and} \,\,(n_b(w) >= 5)\}$
As $L$ and $\bar{L}$ require same number of states, so we will figure out the number of states for $\bar{L}$ and the same will hold for $L$ as well.
Essentially $\bar{L}$ has two conditions: first being divisibility on the count of $a$'s by 3, and second on count of $b$'s being at least 5. Hence the resulting product automata requires $(3 \times 5) = 15$ states.
