When, recently, I did a conversion of a positive integral Celsius temperature, C = 275, to its Fahrenheit equivalent, it turned out to be F = 527, and I notice that I could have simply moved the digit at right of C, to the front, to obtain F. After some intense calculations, I failed to discover the next largest such example.

Does one exist, and if so, what is it?

Suppose the solution is abc...z, n digits long, and let's write the continued fraction C=0.abc...zabc...z..... Then, F=0.zabc...zabc...z... = (C+z)/10. But, according to the centigrade/fahrenheit formula, F=Cx(9/5)+0.00...3200...32... = Cx(9/5)+32/(10^n-1).<p>
Equating formulae, we get z(10^n-1)-320=(10^n-1)17C, so we need z(10^n-1)-320 to be a multiple of 17.<p>Using some basic theorems, n must be 2+16K, for integer K, and z=5.<p>So, the first answer is (5x999-320)/(17x999)= 0.275 275 275 ..., and the second answer is as Charlie found.

Posted by Federico Kereki on 2005-09-01 00:25:30