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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?

 Jul 11, 2019
 #1
avatar+102447 
+1

Present your question properly.

 
 Jul 11, 2019
 #2
avatar+101813 
+2

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

 

 

 

cool cool cool

 
 Jul 11, 2019
 #3
avatar+625 
+3

Is this an AoPS HW problem? That's a no-no.

 
 Jul 11, 2019
 #4
avatar+102447 
+1

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.

 
 Jul 11, 2019
 #5
avatar+625 
0

what? me? what did I do? or are you talking to the guest? I'm not sure...

 
tommarvoloriddle  Jul 11, 2019
 #6
avatar+102447 
+1

Sorry for the confusion Tom.

I was not referring to you.

 
Melody  Jul 11, 2019
 #7
avatar+625 
0

:) Thank you Melody.

 
tommarvoloriddle  Jul 11, 2019
 #8
avatar+625 
0

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.

 
 Jul 11, 2019
edited by tommarvoloriddle  Jul 11, 2019
 #9
avatar+102447 
0

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.

 
Melody  Jul 11, 2019
 #10
avatar+367 
+3

as of such \(\textcolor{blue}{\bf\circledast}\)

 
travisio  Jul 11, 2019
 #11
avatar+625 
0

: ) ok melody and traviso, thank you

 
tommarvoloriddle  Jul 11, 2019
edited by tommarvoloriddle  Jul 11, 2019

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