Which decimal digits occur as the final digit of a forth power of an integer?

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- Feb 25th 2008, 12:28 PMmandy123decimal digit as final digit
Which decimal digits occur as the final digit of a forth power of an integer?

- Feb 25th 2008, 05:58 PMThePerfectHacker
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.

- Feb 25th 2008, 07:18 PMtopsquark
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