Thea has a key on her calculator marked $\textcolor{blue}{\bf\circledast}$. If an integer is displayed, pressing the $\textcolor{blue}{\bf\circledast}$ key chops off the first digit and moves it to the end. For example, if $6138$ is on the screen, then pressing the $\textcolor{blue}{\bf\circledast}$ key changes the display to $1386$. Thea enters a positive integer into her calculator, then squares it, then presses the $\textcolor{blue}{\bf\circledast}$ key, then squares the result, then presses the $\textcolor{blue}{\bf\circledast}$ key again. After all these steps, the calculator displays $243$. What number did Thea originally enter?
+129899 Working backwards....the last number displayed before pressing the \(\textcolor{blue}{\bf\circledast}\) key would have been 324
Taking the square root of this we get 18
So...before pressing the \(\textcolor{blue}{\bf\circledast}\) key we would have had 81
Taking the square root of this = 9 = the original integer
+118687 Since you could not be bothered writing your question legibly, I wonder if you can be bothered copying CPhill's answer.
But then maybe you only have to click and paste. It is not like you have to understand or write anything for yourself.
Maybe you can manage.
+1713 what? me? what did I do? or are you talking to the guest? I'm not sure...
+1713 The only reason why I guessed this was a hw homework problem was the extra latex, and it had dollar signs. Only AoPS and a few other math sites use that- and they have hw too...
And the DUE soon thing.
+118687 All that funny writing is just Latex that is not run in a latex program.
A lot of maths sites make anything between $ signs automatically be recognised as Latex.
You can set you computer, when seeing things on the internet, to do that too.
This web2 site does not do that though and that is why it looks 'silly' here.
