Abstract Algebra - 4.2 Cyclic Groups and Their Properties a^k=a^gcd(n,k)

In this video we look at the second theorem regarding cyclic groups. This theorem can be useful when finding the generators of a set. I did not review the definition of cyclic groups, so if you aren's clear on what a cyclic group is, please first watch video 4.1. As in the last video, I've done my best to provide an example of when each corollary is used. If you have others, please feel free to drop them in the comments.

Video Chapters:
Intro 0:00
The Property a^k=a^gcd(n,k) 0:23
Consequences of the Property - The First Two Corollaries 3:43
Consequences of the Property - The Last Two Corollaries 7:45
Up Next 13:19

This playlist follows Gallian text, Contemporary Abstract Algebra, 9e.

[anangelsdiaries]

For the last question, shouldn't it be everything except for multiple of 5 instead of 5 itself?
For example gcd(25, 10) would be 5 which would break the corollary. As proof unless I made a mistake <10> would give {10, 20, 5, 15} under addition mod 25.