1999 AMC 8 Problems/Problem 24
When is divided by , the remainder is
Note that the units digits of the powers of 9 have a pattern:,,,, and so on. Since all natural numbers with the same last digit have the same remainder when divided by 5, the entire number doesn't matter, just the last digit. For even powers of , the number ends in a . Since the exponent is even, the final digit is . Note that all natural numbers that end in have a remainder of when divided by . So, our answer is .