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?

Pungce Oct 12, 2018

#1**0 **

Its sure 243 i don't find answer but look at this maybe help you:

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!

Dimitristhym Oct 12, 2018

#3**0 **

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?

Melody Oct 13, 2018