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Math Help - Large Number

  1. #1
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    Large Number

    What is the last (rightmost) digit of

    23^{23^{23^{23}}}?
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  2. #2
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    The ones digits of powers of 23 cycle as do the ones digits of powers of 3, and as 3x3=9, 3x9=27, 3x7=21, 3x1=3, we see that the ones digits cycle through the values 3, 9, 7, 1. Thus, to find the ones digit of 23^{23^{23^{23}}}, we need only find 23^{23^{23}} mod 4. Now, 23 mod 4 = 3, and so we see that
    23^2 mod 4 = 1, and thus 23^n mod 4 is 3 when n is odd and 1 when n is even. And we see that since 23 is odd, 23^{23} is odd, and so 23^{23^{23}} mod 4 = 3, and thus the ones digit of 23^{23^{23^{23}}} is the same as that of 3^3, which is 7.

    --Kevin C.
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  3. #3
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    Lovely.
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