Check out this:
Cyclic group - Wikipedia, the free encyclopedia
and look at the property regarding prime numbers for the order of the groups
First of all, note that cyclic groups are of necessity abelian. So ANY subgroup is automatically normal.
Since G is cyclic, there is some element g of G that generates G. I claim gN generates G/N.
For suppose xN is ANY element of G/N. Since x is some element of G, we have , for some integer k (since G is cyclic, with generator g).
Thus: , which shows that G/N is generated by gN.