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 \[ 5\cdot (10^{19}-65)/17 = 2941176470588235275.\]

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