Question 252455
There is a pattern for the 3's

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81.
3^5 = 243
Notice the pattern {3,9,7,1} repeats after every fourth group. The sum of 3,9,7,and 1 is 20. 2006 / 4 will get you the number of groups. That quotient is 501 remainder 2.

20*501 = 10020. This is the sum of the first 501 groups. Remainder 2 means add 3+9, or 3^2005 + 3^2006 = 3^1 + 3^2 in terms of units digits.

So, 20*501 + 12 = 10032. Don't forget to add back in 3^0 = 1. 
The units digit is 2+1 = 3.