Lets say you have a number like $\displaystyle 2^{1000} $. Then the last two digits of this number is the remainder of $\displaystyle \frac{2^{1000}}{100} $ in $\displaystyle \mathbb{Z}/100 \mathbb{Z} $. And $\displaystyle \mathbb{Z}/100 \mathbb{Z} $ contains $\displaystyle 100 $ residue classes.

Is it okay to generalize, and say that the last $\displaystyle n $ digits of a number $\displaystyle k^{p} $ where $\displaystyle k > 0 $ and $\displaystyle n \leq p $ is the remainder of $\displaystyle \frac{k^{p}}{n} $ in $\displaystyle \mathbb{Z}/n \mathbb{Z} $? And the last $\displaystyle n $ digits are in one of the $\displaystyle n$ residue classes?