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# I really need help

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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 138.

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?

Oct 12, 2018
edited by Pungce  Oct 13, 2018
edited by Pungce  Oct 13, 2018

#1
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for example 6138 = abcd a=6,b=1,c=3,d=8 if we dont know the numbers so:

we have x it's the  originally enter

x -> x^2 -> abcd (the result) -> press a -> bcd ->(bcd)^2 -> efghij -> press e ->fghij BUT fghij=243 SO fghij-> fgh with f=2,g=4,h=3 and e=1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9.The biggest is e=9 so efgh=9243 But sqrt(9243)=96.1 so (bcd)^2 its bc NOT 3 numbers and a=1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 the biggest is 9 But sqrt(999)=32 SO

x<32

and we have finally:

x -> x^2 -> abc -> press a -> bc -> (bc)^2 -> efgh -> press e -> fgh but no one for numbers 1423,2243,3243,4243,5243,6243,7243,8243,9243 dont have integer result for this reason i ask you if its 243 for sure!

I hope I helped you!

Oct 12, 2018
#2
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Cphil can u help me?

Oct 13, 2018
#3
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Thea has a key on her calculator marked . If an integer is displayed, pressing the  key chops off the first digit and moves it to the end. For example, if 6138 is on the screen, then pressing the  key changes the display to 138.

Wouldn't   6138  become  1386 ?    You said the first digit gets moved to the end ....

Thea enters a positive integer into her calculator, then squares it, then presses the  key, then squares the result, then presses the  key again. After all these steps, the calculator displays 243. What number did Thea originally enter?

Oct 13, 2018