I'm work through Gallian's Contemporary Abstract Algebra for my Groups and Symmetries class however I've been stuck trying to understand the following section for awhile.
The theorem is: Let a be an element of order n in a group and let k be a positive integer. Then: $$<a^k> = <a^{gcd(n,k)}> \text{and } |a^k| = \frac{n}{gcd(n,k)}$$
Such that n = order of the element.
So the author proceeds to give an example such that: For |a| = 30. We find,
$$ <a^{26}>, <a^{17}>, <a^{18}> $$
I have no idea where he gets these generators from. I understand the arithmetic using them to find other generators that give the same subgroup however no clue on where he gets them from.
Any clarification would be greatly appreciated.
Here is the full example
