Show that C(n) = {a^k | k is a multiple of n} is a regular language I came across this question in an exam book and was unable to find a solution:

Prove that C(n) = {a^k | k is a multiple of n} is a regular language for every natural number n ≥ 1.

I wasn't able to construct a DFA as the number n is left variable in the original question. Is it possible to construct a DFA with a variable number of states?

Any help would be greatly appreciated.
Thank you!
It’s easy to construct a DFA which recognize $a^n$. 

Then add an $\epsilon$ transition from all its final state to its initial state.