- #1

Math100

- 771

- 219

- Homework Statement
- For any integer ## a ##, verify that ## a^{5} ## and ## a ## have the same units digit.

- Relevant Equations
- None.

Proof:

Let ## a ## be any integer.

Applying the Fermat's theorem produces:

## a^{2}\equiv a\pmod {2}, a^{5}\equiv a\pmod {5} ##.

Observe that

\begin{align*}

&a^{4}\equiv a^{2}\pmod {2}\equiv a\pmod {2}\\

&a^{5}\equiv a^{2}\pmod {2}\equiv a\pmod {2}.\\

\end{align*}

This means ## a^{5}\equiv a\pmod {10} ##.

Suppose ## 0\leq k<10 ##.

Then ## a^{5}-k\equiv (a-k)\pmod {10} ##.

Thus ## a^{5}-k\equiv 0\pmod {10}\implies a-k\equiv 0\pmod {10} ##.

Therefore, ## a^{5} ## and ## a ## have the same units digit for any integer ## a ##.

Let ## a ## be any integer.

Applying the Fermat's theorem produces:

## a^{2}\equiv a\pmod {2}, a^{5}\equiv a\pmod {5} ##.

Observe that

\begin{align*}

&a^{4}\equiv a^{2}\pmod {2}\equiv a\pmod {2}\\

&a^{5}\equiv a^{2}\pmod {2}\equiv a\pmod {2}.\\

\end{align*}

This means ## a^{5}\equiv a\pmod {10} ##.

Suppose ## 0\leq k<10 ##.

Then ## a^{5}-k\equiv (a-k)\pmod {10} ##.

Thus ## a^{5}-k\equiv 0\pmod {10}\implies a-k\equiv 0\pmod {10} ##.

Therefore, ## a^{5} ## and ## a ## have the same units digit for any integer ## a ##.