Find the last digit of $3^{1006}$ The way I usually do is to observe the last digit of $3^1$, $3^2$,... and find the loop. Then we divide $1006$ by the loop and see what's the remainder. Is it the best way to solve this question? What if the base number is large? Like $33^{1006}$? Though we can break $33$ into $3 \times 11$, the exponent of $11$ is still hard to calculate.

You have 
\begin{cases}3^1& =3\\ 3^2& =9 \\ 3^3&=27\\3^4&= 81\\3^5&= 243\end{cases} Thus the last digit repeats after $4$ steps. Since $1006=251\cdot 4+2$ it is
$$3^{1006}=(3^4)^{251}\cdot 3^2.$$ The last digit of $(3^4)^{251}$ is $1$ and the last digit of $3^2=9.$ So the answer you are looking for is $9.$ 
To compute the last digit of $33^{1006}$ you are in the right way. Since $33=3\cdot 11$ and the last digit of $11^n$ is $1$ you have that the last digit of $33^{1006}$ is the same of $3^{1006}.$
