Automata question: Find example of language L with p = n = m = k

[paul2834]

The question:
Let it be $\displaystyle L$ a regular language.
few definitions:
$\displaystyle p(L)$-the minimal natural number so that $\displaystyle L$ fulfill the pumping lemma.
$\displaystyle n(L)$- minimal NFA that accepts $\displaystyle L$.
$\displaystyle m(L)$- $\displaystyle Rank(L)$, the number of equivalence classes in $\displaystyle L$.

Let it be integer $\displaystyle k>0$. find an example for a language $\displaystyle L$ so that: $\displaystyle p=n=m=k$.


My attempt:
At start, thought of $\displaystyle L=\{w: |w|\bmod k=0\}, \Sigma=\{ a\}$.
But then I realized that $\displaystyle p<k$.
And also- for every $\displaystyle L$ that I choose here, I don't know how exactly to prove that the $\displaystyle NFA$ which accepts it is minimal. I know how to do this only on $\displaystyle NFA$'s with one or two states, but not on $\displaystyle NFA$'s with unknown number of states.
How can I solve this?