第五届华中区程序设计邀请赛暨武汉大学第十四届校赛 现场赛 WHU oj Problem 1613 - Count in Sama’s triangle 扩展欧几里得

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Time Limit: 1000MS   Memory Limit: 65536KB   

Description
        Today, the math teacher taught Alice Hui Yang’s triangle. However, the teacher came up with a new one, called Sama’s triangle (some of you may be familiar with this name).

       In this triangle, there is only 30 row, and the ith row has i-1 number, f[i][1] , f[i][2], …, f[i][i-1] (1strow has no number). And initially, f[i][0] = a, f[i][i] = b, (0        2ndrow    f[2][1]
       3rdrow     f[3][1]     f[3][2]
       4throw     f[4][1]     f[4][2]     f[4][3]
       30throw   f[30][1]   f[30][2]   f[30][3]   …    f[30][29]
       Now the teacher asked, ‘how many M are in the triangles among all pairs of (a, b).
Input
The input consists of several test cases.
The first line consists of one integer T (T <= 500), meaning the number of test cases.
For each test cases, there is one integer M (M<= 1e9), which is asked from the teacher.
Here, 1eX means 1000…0000 (x 0s).
Output
For each test case, output the times number M shows up in all positions of all triangles.
Sample Input
1
3
Sample Output
4

题意：

给出30*30的矩阵 其中j>=i的矩阵部分不要

令 f[i][0] = a, f[i][i] = b, (0

给出一个m问  对于任意的(a,b)能产生多少个m  这里的意思是多少个位置存在m 对于所有的(a,b)组合

思路：

推出三角矩阵发现所有的位置上都是关于一个x*a+y*b (x,y为常数)的一元二次方程  

然后我们需要求出这个位置上的x*a+y*b==m 有多少解  也就是有多少对这样的(a,b)组合

并且每个位置的a,b互不影响  （后来证明了其实也影响不了  因为所有的（x,y）都是不想同的

下面就是求不定方程a*x+b*y==m有多少正整数解  完美利用扩展欧几里得求出贡献即可

这里还需要注意的就是a*x+b*y=m要存在正整数解的话 需要满足 m%gcd(a,b)==0 

代码：

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