<s>[INST] Pattern to last three digits of power of $3$? I'm wondering if there is a pattern to the last three digits of a a power of $3$? I need to find out the last three digits of $3^{27}$, without a calculator.
I've tried to find a pattern but can not see one? Am I missing something?
Thanks for your help in advance! 
 [/INST] Exponentiation by squaring reduces the number of multiplications from 26 to 7:
First square repeatedly to create powers by powers of 2:
$$ 3^1=3 \qquad 3^2=9 \qquad 3^4=81 \qquad
3^8 \equiv 561 \pmod{1000} \qquad
3^{16} \equiv 721 \pmod{1000} $$
Then, since $27=1+2+8+16$,
$$ 3^3=3^1 3^2 = 27 \qquad
3^{11} = 3^3 3^8 \equiv 147 \pmod{1000} \qquad
3^{27} = 3^{11} 3^{16} \equiv 987 \pmod{1000} $$

We can even be smarter, as gt6989b noted in a comment, and use $3^{27} = ((3^3)^3)^3$ and get down to 6 multiplications:
$$ 3^2 = 9 \qquad 3^3 = 27 $$
$$ (3^3)^2 = 729 \qquad 3^9 = (3^3)^3 \equiv 683 $$
$$ (3^9)^2 \equiv 489 \qquad 3^{27} = (3^9)^3 \equiv 987 $$
But in general the trouble of looking for such tricks is not really worth it over just using plain squaring.
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