Monday, June 24, 2013

Twisty Temperature

Problem:

When Waldo recently did a conversion of a positive integral Celsius temperature $$c = 275$$ to its Fahrenheit equivalent $$f$$ (which turned out to be $$527$$ ), he noticed to his amazement that he could have simply moved the last digit of $$c$$ to the front to obtain $$f$$. Doing some intense calculations he failed to discover the next largest such example. Does one exist, and if so, what is it?

Solution:

Let $$c = x_{n}\cdot 10^{n-1} + ... + x_{1}\cdot 10^1 + x_{0}$$ with $$x_{n} > 0$$, then $f = x_{0}\cdot 10^{n-1} + (c-x_{0})/10.$ We also have that $f = (9/5)\cdot c + 32.$ Notice that in order for $$f$$ to be integral $$c$$ must be divisible by 5; this implies that $$x_0=5$$ since it cannot equal 0 (since as a number $$f>c$$). Our equation then becomes $(9/5)\cdot c + 32 = 5\cdot 10^{n-1} + (c-5)/10$ implies $c = 5\cdot (10^n - 65)/17.$ Now it turns out that 10 is a primitive root modulo 17, and so it follows that $$c$$ is integral if and only if $$n$$ is of the form $$16\cdot m + 3$$. When $$m=0$$ we get $$c=275$$; when $$m=1$$ we get the next highest such temperature, which is $5\cdot (10^{19}-65)/17 = 2941176470588235275.$