Definition of a language $L$ with alphabet $\{a\}$ is given as following. $$L = \{a^{nk} \mid k > 0, \text{ and } n \text{ is a positive integer constant}\}$$ What is the minimum number of states needed in a DFA to recognize $L$?

The language is defined as:

Here, $n$ is a fixed positive integer constant, and $k \ge 1$.

• The string $\epsilon$ (length $0$) is not in $L$ because $k > 0$.

• The initial state $q_0$ corresponds to length $0$ and must be a non-accepting state.

• From $q_0$, reading $1, 2, \dots, n-1$ copies of '$a$' brings the DFA to non-accepting states $q_1, q_2, \dots, q_{n-1}$.

• Reading the $n$-th '$a$' leads to state $q_n$, which is an accepting state.

• Subsequent transitions cycle through $q_1, q_2, \dots, q_n$ because reading another '$a$' from $q_n$ transitions to $q_1$.

Since $q_0$ cannot be merged with $q_n$ (as $\epsilon \notin L$), the states are:

• $q_0$ (start state, non-accepting)

• $q_1, q_2, \dots, q_{n-1}$ ($n-1$ non-accepting states)

• $q_n$ (accepting state)

Total number of states required in the minimal DFA = $1 + (n-1) + 1 = n + 1$.

Hence, the correct option is (B).