Originally Posted by

**Soroban** Hello, spywx!

Welcome aboard!

Did you try *anything?*

I would try consecutive powers of 7 and see what happens . . .

$\displaystyle \begin{array}{ccc}\text{power} & & \text{ends in:} \\ \hline

7^1 & \to & 7 \\

7^2 & \to & 9 \\

7^3 & \to & 3 \\

7^4 & \to & 1\\

\vdots & & \vdots \end{array}$

We see that $\displaystyle 7^4$ ends in 1.

We have: .$\displaystyle 7^{1,000,000} \;=\;(7^4)^{250,000} \:\to\:1^{250,000} \:\to\:1 $

Try consecutive powers of 11 . . .

$\displaystyle \begin{array}{ccc}\text{power} & & \text{ends in:} \\ \hline

11^1 & \to & 1 \\

11^2 & \to & 1 \\

11^3 & \to & 1 \\

& \text{Hey!} \end{array}$