13^655: Find the last digit of a number raised to a large power

$13^1 = 13$

$13^2 = 169$

$13^3 = 2,197$

$13^4 = 28,561$

$13^5 = 371,293$

$13^6 = 4,826,809$

$13^7 = 62,748,517$

$13^8 = 815,730,721$

$13^9 = 10,604,499,373$

$13^{10} = 137,858,491,849$

$13^{11} = 1,792,160,394,037$

$13^{12} = 23,298,085,122,481$
 

Notice the last digit of the number in each row is making the following pattern
$$ 3, ~ 9, ~ 7, ~ 1, ~ 3, ~ 9, ~ 7, ~ 1, ~ 3, ~ 9, ~ 7, ~ 1 $$

The pattern 3, 9, 7, 1 repeats every four rows. Now count how many times this pattern will repeat in 655 rows. To do that, simply count how many 4-rows are there in 655.
$$ 655 \div 4 = 163 ~ \text{remainder} ~ 3$$

This simply means that there are 163 counts of full { 3, 9, 7, 1 } and the remainder represents 3 more rows. The three remaining rows in the pattern are { 3, 9, 7 }.

Thus, the last digit of 13⁶⁵⁵ is $7$.