a) Let $|a| = 40$. How many left cosets of $\langle a^6 \rangle$ in $\langle a \rangle$ are there?
b) Let $|a| = n$. How many left cosets of $\langle a^k \rangle$ in $\langle a \rangle$ are there?
For part a, I managed to list out all of the elements of $\langle a^6 \rangle$. Taking the identity $\epsilon$ and doing $\epsilon \langle a^6 \rangle$, I get the first coset and there turned out to be $20$ elements. So by the property that all cosets of a subgroup have the same cardinality, two cosets are either equal or disjoint, and each element appears in exactly one coset, I concluded that there are $2$ left cosets of $\langle a^6 \rangle$ in $\langle a \rangle$.
I am not sure how to show part b. Is there any connection from part a to b? Any help would be great, thanks.
Also, I have not yet learned Lagrange's theorem.