Units Digits of k and k^5 always the same: why?

[Trenters4325]

Why are the units digits of k and k^5 always the same?

 

[stapel]

What is 0⁵? What is 1⁵? What is 2⁵? 3⁵? 4⁵? On up to 9⁵?

Then, whatever k might be, whatever units digit it might have, how will this compare with k⁵?

Eliz.

 

[soroban]

Re: Units Digits

Hello, Trenters4325!

Why are the units digits of $\displaystyle k$ and $\displaystyle k^5$ always the same?

Good question! . . . I had noticed that, too.

Of course, it depends on the base of the number system being used.

The units digits of $\displaystyle k$ and $\displaystyle k^5$ are the same in base-ten.

I don't have anything near a proof, but I've made some observations
$\displaystyle \;\;$and here are my conjectures . . .

This phemenon occurs because $\displaystyle 5$ is exactly half the base (10),
$\displaystyle \;\;$but it doesn't work for all even bases.
It seems that the base must be twice an odd prime.

In base-6:
$\displaystyle \;\;\;\begin{array}{cccccc}0^3\,\to\,0 \\ 1^3\,\to\,1 \\ 2^3\,\to\,2\\3^3\,\to\,3\\4^3\,\to\,4 \\ 5^3\,\to\,5\end{array}$

In base-14:
$\displaystyle \;\;\;\begin{array}{ccccccc}1^7\,\to\,1\\2^7\,\to\,2\\3^7\,\to\,3\\4^7\,\to\,4\\5^7\,\to\,5\\\vdots\\13^7\,\to\,13\end{array}$


But, like I said, I'm only guessing.
Perhaps someone else can enlighten us . . .