Question

What is the ones digit when you do 9 to the power of 999?

Answer 1

9

Explanation:

9 is congruent to -1 mod 10

(-1)^999 = -1, because 999 is odd, and when you raise -1 to an odd power you get -1

so the last digit is congruent to -1 mod 10

hence the last digit is 9

Answer 2

9

Explanation:

You want the 1s digit of 9^999.

Ones digit

The first few powers of 9 are ...

9^1 = 9

9^2 = 81

9^3 = 729

The ones digit of the power alternates between 9 and 1. It is 1 for even powers of 9, and 9 for odd powers of 9.

9^999 is an odd power of 9, so its ones digit is 9.

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