I'd really love your help with the following problem. I'm trying to use couple of theorems that I know, but I'm not sure if I'm allowed to and if I'm doing it correctly.
The question is simple: I need to find all the four primitive roots of modulo 26 and the eight primitive roots modulo 25, it's just that I'm kind of lost with what to use or to do in this case.
This is what I tried to do: 26 is not a prime number. $26=2^1 \cdot 13^1$, so $u(\mathbb{Z}/_{26}\mathbb{Z})\simeq u(\mathbb{Z}/_{2}\mathbb{Z})u\times (\mathbb{Z}/_{13}\mathbb{Z})$, I already know that the primitive roots of 13 are $2,6,7,11$.
$(a)$What does $u(\mathbb{Z}/_{26}\mathbb{Z})\simeq u(\mathbb{Z}/_{2}\mathbb{Z})u\times (\mathbb{Z}/_{13}\mathbb{Z})$ really mean? and $(b)$ how does it help me find all the primitive roots of modulo 26 once I already know $13$s?
For 25, there's a theorem saying that $g^k$ is a primitive root modulo m if and only if $(k, \phi (m)) = 1$, for a primitive root modulo $m$. since I already know that $2$ is a primitive root modulo $25$, I can use this theorem.$(c)$ Can I use the same method for $26$?
Thanks a lot!