decimal expansion?

Question

Find the last digit of the decimal expansion of 3^(1000)

Answer 1

3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27 last digit 7
3^4 = 81 last digit 1
4-cycle, so divide 1000 by 4, get remainder 0, so 3^1000 ends in 1.

Answer 2

3^2 = 9
3^3 = 27
3^4 = 81

thus the last digits go by the sequence
3 - 9 - 7 - 1

thus
3^(1000) has a last digit of 1... since 1000 is divisible by 4.

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