Finding The Order of Elements This is a homework problem from my Group Theory class.
What is the order of $6$ in $\mathbb Z_{16}$?
I know $\mathbb Z_{16} = \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$.
I know that in order to find the $\operatorname{ord}(6)$ I need to find $n,$ such that $6^n=e,$ where $e$ is the identity.
My main problems/questions here are:
Is the operation on $\mathbb Z_{16}$ addition [modular to be exact]?
If so, $e=0$. 
$$6+6=12+6=2+6=8+6=14+6=4+6=10+6=0$$
Therefore $6^8=0.$
I am not sure if my work is correct.  And if it is, is there a different way of going about it? For instance if I am trying to find the $\operatorname{ord}(6) \in \mathbb Z_{100}$ instead...or any larger group. 
Thank you!

You've got it! 
You've found the order of $6 \in \mathbb Z_{16}$:  the additive group of integers, modulo $16$. $$|\langle 6\rangle| = 8$$
Once you understand that the order of an element $g$ in a finite group is given by the least positive integer $n \neq 0$ such that $g^n = e$, or in an additive group, the least positive integer $n \neq 0$ such that $ng = 0$, and compute a few cycles like you did above: 

Then you can more quickly use the fact that for the additive group of integers modulo $n$, the order of an element $g$ in $\mathbb Z_n$ is given by $$\operatorname{ord}(g) = \dfrac{n}{\gcd(g,n)}$$

