TITLE: Pattern to last three digits of power of $3$?
QUESTION [13 upvotes]: 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!

REPLY [2 votes]: If we generate the list, and format it appropriately, some patterns immediately become apparent:
001, 003, 009, 027, 
081, 243, 729, 187, 
561, 683, 049, 147, 
441, 323, 969, 907, 
721, 163, 489, 467, 
401, 203, 609, 827, 
481, 443, 329, 987, 
961, 883, 649, 947, 
841, 523, 569, 707, 
121, 363, 089, 267, 
801, 403, 209, 627, 
881, 643, 929, 787, 
361, 083, 024, 747, 
241, 723, 169, 507, 
521, 563, 689, 067, 
201, 603, 809, 427, 
281, 843, 529, 587, 
761, 283, 849, 547, 
641, 923, 769, 307, 
921, 763, 289, 867, 
601, 803, 409, 227, 
681, 043, 129, 387, 
161, 483, 449, 347, 
041, 123, 369, 107, 
321, 963, 889, 667

The units are always odd (obviously), and increase by $2, 6, 8, 4$ 
The 10s in each column is increasing by the same amount on each row (again, $8, 4, 2, 6$)
The only place a pattern isn't immediately spottable is in the 100s digit. If you check carefully though, you can see a recurrence every 5 lines (adding $4, 2, 6, 8$ to the previous set of 5 lines): (0, 0, 5, 4, 7) + 4 = (4, 4, 9, 8, 1).
Interestingly, $2, 4, 8, 6$ are the only values of $2^n mod 10, n>0$.  There's some significance there, but it's been a while since I studied number theory.