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Celsius x Fahrenheit (Posted on 2005-08-31) Difficulty: 2 of 5
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?

  Submitted by pcbouhid    
Rating: 3.3333 (3 votes)
Solution: (Hide)

Let C = x(0) + x(1)*10^1 + x(2)*10^2 +...+ x(n)*10^(n-1). In other words, x(0), x(1), x(2), ..., x(n) are the digits of the number C, from right to left.

Then, F = [C - x(0)]/10 + x(0)*10^(n-1).

We also know that to convert a temperature from the Celsius scale to the Fahrenheit scale, the formula is :

F = (9/5)*C + 32.

Then, in order for F to be integral, C must be divisible by 5, and this implies that x(0) = 5, since it cannot be zero (F is greater than C).

We have, then :

(9/5)*C + 32 = (C - 5)/10 + 5*10^(n-1), or :

C = 5*(10^n - 65)/17.

17 * C = 5*10^n - 325

C is congruent to 5*10^n - 2 (mod 17)

The sixteen first powers of 10 (mod 17), could be easyly evaluated, and they are, in order, congruent to (10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 0, 8, 12, 1).

Multiplying these numbers by 5 and subtracting 2, we'll find that (5*10^n - 2) is congruent to 0 (mod 17), only when n is equal to 3 (5*14 - 2 = 68 = 4*17), or generalizing, equal to 16*m + 3.

For m = 0, we have n = 3, which give us C = 5*(10^3 - 65)/17 = 275.

The next number is obtained with m = 1.....n = 19.....that give us C = 5*(10^19 - 65)/17, which is

C = 2,941,176,470,588,235,275.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
answerK Sengupta2008-02-26 12:08:57
SolutionWithout a computerFederico Kereki2005-09-01 00:25:30
How about without computer?pcbouhid2005-08-31 21:16:03
re: Some Big Numbers -- Computer SolutionBob Smith2005-08-31 20:02:36
SolutionSome Big Numbers -- Computer SolutionCharlie2005-08-31 19:50:15
re(2): I think I found one! (I didn't)Charlie2005-08-31 19:34:59
re: I think I found one! (I didn't)Bob Smith2005-08-31 18:50:14
re: I think I found one!pcbouhid2005-08-31 18:48:06
Some ThoughtsI think I found one!Bob Smith2005-08-31 18:41:11
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