elementary number theory - 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.

Answer

You have $$3^2=9\equiv -1\pmod{10}.$$

And $1006=503\times 2$, so

$$3^{1006}=(3^2)^{503}\equiv (-1)^{503}\equiv -1\equiv 9\pmod{10}.$$

So the last digit is $9$.

---

And for something like $11$, you can use the fact that $11\equiv 1\pmod {10}.$