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Perfect Powered Temperature (Posted on 2022-01-09) Difficulty: 3 of 5
Each of M and N is a nonnegative integer such that:

Mo Celsius = No Fahrenheit.

Determine all possible values of N<10 such that N is a perfect power.

*** F = (9/5)*C +32, where F denotes degree Fahrenheit and C denotes degree Celsius.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 2 of 8 |
I think there must be a typo in the problem, because if M >= 0 and N >= 0 there are no such integers, let alone perfect powers.  For any Fahrenheit temp below 32, the Celsius value is negative.
So I tested Celsius values from -17 up to +10^7 degrees C.

<table cellspacing="0" border="0"> <colgroup span="2" width="85"></colgroup> <colgroup span="2" width="68"></colgroup> <tbody><tr> <td height="17" align="center">Celsius</td> <td align="center">Fahrenheit</td> <td align="center">base</td> <td align="center">power</td> </tr> <tr> <td height="17" align="center" sdval="0" sdnum="1033;0;#,##0">0</td> <td align="center" sdval="32" sdnum="1033;0;#,##0">32</td> <td align="center" sdval="2" sdnum="1033;">2</td> <td align="center" sdval="5" sdnum="1033;">5</td> </tr> <tr> <td height="17" align="center" sdval="1120" sdnum="1033;0;#,##0">1,120</td> <td align="center" sdval="2048" sdnum="1033;0;#,##0">2,048</td> <td align="center" sdval="2" sdnum="1033;">2</td> <td align="center" sdval="11" sdnum="1033;">11</td> </tr> <tr> <td height="17" align="center" sdval="43385" sdnum="1033;0;#,##0">43,385</td> <td align="center" sdval="78125" sdnum="1033;0;#,##0">78,125</td> <td align="center" sdval="5" sdnum="1033;">5</td> <td align="center" sdval="7" sdnum="1033;">7</td> </tr> <tr> <td height="17" align="center" sdval="72800" sdnum="1033;0;#,##0">72,800</td> <td align="center" sdval="131072" sdnum="1033;0;#,##0">131,072</td> <td align="center" sdval="2" sdnum="1033;">2</td> <td align="center" sdval="17" sdnum="1033;">17</td> </tr> <tr> <td height="17" align="center" sdval="89455" sdnum="1033;0;#,##0">89,455</td> <td align="center" sdval="161051" sdnum="1033;0;#,##0">161,051</td> <td align="center" sdval="11" sdnum="1033;">11</td> <td align="center" sdval="5" sdnum="1033;">5</td> </tr> <tr> <td height="17" align="center" sdval="4660320" sdnum="1033;0;#,##0">4,660,320</td> <td align="center" sdval="8388608" sdnum="1033;0;#,##0">8,388,608</td> <td align="center" sdval="2" sdnum="1033;">2</td> <td align="center" sdval="23" sdnum="1033;">23</td> </tr> </tbody></table>

  Posted by Larry on 2022-01-09 10:17:05
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