A language is a SET of strings.
A relation is defined on a set which relates one element to another. Here, set is $\Sigma^*$ and we have $L$ a subset of it.
Now, given a set of strings I define a relation $R$ such that $aRb$ ($a$ and $b$ elements of $\Sigma^*$) if and only if for any string $z \in \Sigma^*,$ $az \in L \implies bz \in L$ and $az \notin L \implies bz \notin L$ (double implication). Now, this relation is an equivalence relation (see for reflexivity, symmetry and transitivity). Here, if we get any $z \in \Sigma^*,$ such that $az \in L$ and $bz \notin L$ or vice verse $z$ is called the distinguishing string and $a$ and $b$ goes to different equivalence classes (not related).
Now, how many equivalence classes can we get from a given $L$? - No easy way, but can be done with some effort. If this number is finite, then $L$ is a regular language. Further, this number also equals the number of states in the minimal DFA for $L$.
Now for the given $L$, assume $n=2$ (any other positive value for $n$ is fine, but only one value is there for $n$, whereas $k$ takes all positive values). Thus,
$L = \{aa, aaaa, aaaaaa, aaaaaaaa, \dots \}$.
$\Sigma^* = \{\epsilon, a, aa, aaa, aaaa, \dots \}$
$\epsilon$ related to $a$ ? No, because for $z = a,$ $\epsilon z = a \notin L, $where as $az = aa \in L$.
Likewise if we do, we can see that
$\epsilon$ forms an equivalence class - this is not in $L$ but if we append "aa" we get to $L$.
$a,aaa,aaaa,\dots$ forms another equivalence class - these are not in $L$, but if we append "a" to any of these, we get to $L$.
$aa, aaaa,aaaaaa,\dots$ forms another equivalence class - these are in $L$, if we append "aa" to any of these, we remain in $L$.
No other string remain in $\Sigma^*$as we considered all odd length strings and even length strings and also 0 length string by now. So, we got 3 equivalence class. If we see in general for $n$ it will be $n+1$ equivalence class here. A good way in GATE for these questions will be to assume $n=2$, and try a minimal DFA rather than going for equivalence classes.
https://gateoverflow.in/723/gate2001-2-5
