How do you determine the order a cyclic subgroup?
Specifically when dealing with a certain number like $12$?
I know that the order of a subgroup is simply put, the number of distinct elements in the subgroup.
My confusion is what does it mean to generate elements by $12$?
How many times do you need to sum $12$ with itself to get $0 \pmod {84}$? That's the order of $12$ in $\mathbb{Z}_{84}$. Why $0$? Because that's the unit of the additive group $\mathbb{Z}_n$, whatever $n$.