巧妙利用中國剩餘定理(九度 1142)

題目描述：

    Some people believe that there are three cycles in a person's life that start the day he or she is born. These three cycles are the physical, emotional, and intellectual cycles, and they have periods of lengths 23, 28, and 33 days, respectively. There is
one peak in each period of a cycle. At the peak of a cycle, a person performs at his or her best in the corresponding field (physical, emotional or mental). For example, if it is the mental curve, thought processes will be sharper and concentration will be
easier. 
Since the three cycles have different periods, the peaks of the three cycles generally occur at different times. We would like to determine when a triple peak occurs (the peaks of all three cycles occur in the same day) for any person. For each cycle, you will
be given the number of days from the beginning of the current year at which one of its peaks (not necessarily the first) occurs. You will also be given a date expressed as the number of days from the beginning of the current year. You task is to determine
the number of days from the given date to the next triple peak. The given date is not counted. For example, if the given date is 10 and the next triple peak occurs on day 12, the answer is 2, not 3. If a triple peak occurs on the given date, you should give
the number of days to the next occurrence of a triple peak.

輸入：

    You will be given a number of cases. The input for each case consists of one line of four integers p, e, i, and d. The values p, e, and i are the number of days from the beginning of the current year at which the physical, emotional, and intellectual cycles
peak, respectively. The value d is the given date and may be smaller than any of p, e, or i. All values are non-negative and at most 365, and you may assume that a triple peak will occur within 21252 days of the given date. The end of input is indicated by
a line in which p = e = i = d = -1. 

輸出：

    For each test case, print the case number followed by a message indicating the number of days to the next triple peak, in the form:

    Case 1: the next triple peak occurs in 1234 days.

    Use the plural form "days'' even if the answer is 1. 

範例輸入：

0 0 0 00 0 0 1005 20 34 3254 5 6 7283 102 23 320203 301 203 40-1 -1 -1 -1

範例輸出：

Case 1: the next triple peak occurs in 21252 days.Case 2: the next triple peak occurs in 21152 days.Case 3: the next triple peak occurs in 19575 days.Case 4: the next triple peak occurs in 16994 days.Case 5: the next triple peak occurs in 8910 days.Case 6: the next triple peak occurs in 10789 days.

簡單的來說，就是求一個最小的數X。使得 X % 23 = p; X % 28 = e; X % 33 = i;

這就是著名的中國剩餘定理。

數學上的常規解法是這樣的:

找到一個找到一個自然數t，使得  (e*i) * t % 23 = p  . 這裡的t要從1開始試。 令 a = (e*i) * t

再依次找到 b,c。   a+b+c則符合條件。

編程中，有一個更簡便的方法。

找到一個數使得X , 使得 X % (e * i) = 1; （可使用上面的方法一個一個試）。

則 X * p % (e * i) = p; 

代碼如下：

5544 % 23 = 1;  5544 % (28 * 33) = 0;

#include <stdio.h>int main(){int p, e, i, n, sum, t;t = 0;while (scanf("%d%d%d%d", &p, &e, &i, &n), p + e + i + n != -4){sum = (5544 * p + 14421 * e + 1288 * i) % 21252;if (sum - n <= 0)sum += 21252;printf("Case %d: the next triple peak occurs in %d days.\n", ++t,sum - n);}return 0;}