Sequences and Patterns

The units digits of the powers of 3 follow a repeating pattern: 3 , 9 , 2 7 , 8 1 , 24 3 , 72 9 , 218 7 , 656 1 , etc. Thus, The pattern P3 = {3, 9, 7, 1}

The units digits of the powers of 6 follow a repeating pattern: 6 , 3 6 , 21 6 , 129 6 , 777 6 , 4665 6 , 27993 6 , 167961 6 , etc (always 6). Thus, The pattern P6 = {6}

There are 16 repeats of the patterns from $S_1$ to $S_{64}$ , inclusive. The patterns begin again on $S_{65}$ . Thus, $3^{66}$ has the same units digit as $3^2$ , which is 9 , $6^{66}$ has the same units digit as $6^2$ which is 6 . Hence, 9 + 6 - 5 = 10

So, the units digit of $S_{66}$ is 0