R is a rational number such that 2^123456789 degrees Celsius is precisely equal to R degrees Fahrenheit.

(I) Determine the first two digits (reading left to right) in [R].

(II) Determine the last two digits (reading left to right) in [R].

(III) Determine the first digit (reading left to right) following the decimal point in R.

Note: [x] denotes the greatest integer ≤ x.

print 123456789*log(2)/log(10)+log(9/5)/log(10)
 37164196.912631546420776442430889649972415415276
 
Indicates that the required [R] has 37164197 digits when written in decimal, and the first few are

10^.912631546420776442430889649972415415276,

but written without the decimal point. That comes to

81777070058448542420198114358090551277928154067512436....

Of course, we've neglected to add 32, but that comes way at the end, so at some point there's a lack of carry, and we can safely say the first two digits are 81, and in fact, up to the terminal 6 we are safe for however many we show (the 6 itself might really be a 5 if rounding was up to make it a 6, or 7 in the unlikely event that carries percolated up this far).

Posted by Charlie on 2010-12-18 13:35:43