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Thread: decimal digit as final digit

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    decimal digit as final digit

    Which decimal digits occur as the final digit of a forth power of an integer?
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    Reduce $\displaystyle x^{4}$ mod $\displaystyle 10$. You can just reduce $\displaystyle 0^4,1^4,...,9^4$ by mod $\displaystyle 10$ since it will repeat. And those are the possibilities.
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    Quote Originally Posted by mandy123 View Post
    Which decimal digits occur as the final digit of a forth power of an integer?
    Think of a decimal number as
    $\displaystyle a_n \cdot 10^n + ~...~ + a_1 \cdot 10 + a_0$

    So when we raise this to the 4th power we get something like
    $\displaystyle b_{4n} \cdot 10^{4n} + ~...~ + b_1 \cdot 10 + a_0^4$

    So the last digit will be given by the last digit of the original number raised to the fourth power.

    The possibilities are
    $\displaystyle 1 \implies 1$

    $\displaystyle 2 \implies 6$ (16)

    $\displaystyle 3 \implies 1$ (81)

    $\displaystyle 4 \implies 6$ (256)

    etc.

    I get {1, 5, 6}.

    -Dan
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