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.

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