



 
Ryan, What you want to note is that when you take powers of 3 the units digits generate a sequence 3, 9, 7, 1, 3, 9, 7, 1, ...; so for example every fourth one is a 1 and 3^{1000} would end in a 1 as 1000 = 250 x 4. Your problem can be looked at as finding the units digit of 3^{4012} + 3^{2006} (why is that?). From what we've just said above the 3^{4012} must end in a 1 now what about 3^{2006}? Penny
 


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