Celsius x Fahrenheit (Posted on 2005-08-31)

	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.

	Subject	Author	Date
	answer	K Sengupta	2008-02-26 12:08:57
	Without a computer	Federico Kereki	2005-09-01 00:25:30
	How about without computer?	pcbouhid	2005-08-31 21:16:03
	re: Some Big Numbers -- Computer Solution	Bob Smith	2005-08-31 20:02:36
	Some Big Numbers -- Computer Solution	Charlie	2005-08-31 19:50:15
	re(2): I think I found one! (I didn't)	Charlie	2005-08-31 19:34:59
	re: I think I found one! (I didn't)	Bob Smith	2005-08-31 18:50:14
	re: I think I found one!	pcbouhid	2005-08-31 18:48:06
	I think I found one!	Bob Smith	2005-08-31 18:41:11