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 3¹⁰⁰⁰ would end in a 1 as 1000 = 250 x 4.

Your problem can be looked at as finding the units digit of 3⁴⁰¹² + 3²⁰⁰⁶ (why is that?). From what we've just said above the 3⁴⁰¹² must end in a 1 now what about 3²⁰⁰⁶?

Penny