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.

$3^{1006}$ or $33^{1006}$
doesn't really matter
$33\equiv 3\pmod {10}\\
33^{1006}\equiv 3^{1006}\pmod {10}$
$3^4 = 81$
You might say this as $3^4\equiv 1 \pmod{10}$
The last digit of $3^n$ is the same last digit as $3^{n+4k}$
that is:
$(3^{n+4k}) = (3^n)(3^{4k})\equiv (3^n)(1) \pmod{10}$
$1006 = 251\cdot 4 + 2$
the last digit of $3^{1006}$ is the same as the last digit of $3^2$
