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.

HINT:
Find the last digit of $3^{1006\bmod4}$.

Adding a formal solution, just in case anyone will find it useful:
You are looking for $3^{1006}\pmod{10}$.
Since $\gcd(3,10)=1$, by Euler's theorem, $3^{\phi(10)}\equiv1\pmod{10}$.
We know that $\phi(10)=\phi(2\cdot5)=(2-1)\cdot(5-1)=4$.
Therefore $3^{4}\equiv1\pmod{10}$.
Therefore $3^{1006}\equiv3^{4\cdot251+2}\equiv(\color\red{3^4})^{251}\cdot3^2\equiv\color\red{1}^{251}\cdot3^2\equiv1\cdot9\equiv9\pmod{10}$.
