How to check if a number divides a very large number using modulo

Question

How would I determine if 13 divides 12¹² -1

I know the question's asking if 12¹² is congruent to 1 (mod 13), but how would I go about solving this without a calculator?

Answer 1

Hint: $$12\equiv -1\mod 13$$ so $$12^{12}\equiv (-1)^{12}\mod 13$$

Answer 2

Hint: (without the use of congruences)

$$12^{12}-1=(13-1)^{12}-1$$

When expanding the parentheses, how many terms are not divisible by $13$?