HDU 4497 GCD and LCM (数论&组合数学)
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GCD and LCM
http://acm.hdu.edu.cn/showproblem.php?pid=4497
Time Limit: 2000/1000 MS (Java/Others)Memory Limit: 65535/65535 K (Java/Others)
Problem Description
Given two positive integers G and L, could you tell me how many solutions of (x, y, z) there are, satisfying that gcd(x, y, z) = G and lcm(x, y, z) = L?
Note, gcd(x, y, z) means the greatest common divisor of x, y and z, while lcm(x, y, z) means the least common multiple of x, y and z.
Note 2, (1, 2, 3) and (1, 3, 2) are two different solutions.
Note, gcd(x, y, z) means the greatest common divisor of x, y and z, while lcm(x, y, z) means the least common multiple of x, y and z.
Note 2, (1, 2, 3) and (1, 3, 2) are two different solutions.
Input
First line comes an integer T (T <= 12), telling the number of test cases.
The next T lines, each contains two positive 32-bit signed integers, G and L.
It’s guaranteed that each answer will fit in a 32-bit signed integer.
The next T lines, each contains two positive 32-bit signed integers, G and L.
It’s guaranteed that each answer will fit in a 32-bit signed integer.
Output
For each test case, print one line with the number of solutions satisfying the conditions above.
Sample Input
2 6 72 7 33
Sample Output
72 0
Source
2013 ACM-ICPC吉林通化全国邀请赛——题目重现
部分思路来自之前的一道题(UVa 10892题解)。
解释下这里的6是怎么来的:
设L/G=(p1^r1)*(p2^r2)*(p3^r3)…(pm^rm)
又设
x=(p1^i1)*(p2^i2)*(p3^i3)…(pm^im)
y=(p1^j1)*(p2^j2)*(p3^j3)…(pm^jm)
z=(p1^k1)*(p2^k2)*(p3^k3)…(pm^km)
对于某个r,i、j、k里面一定有一个是r,并且一定有一个是0,所以i,j,k有一下3种情况:
r 0 0 ,有C(3,1)种
r 0 r ,有C(3,1)种
r 0 1~r-1 ,有(r-1)*A(3,3)种
所以一共是6*r种。
完整代码:
/*15ms,200KB*/#include<cstdio>int main(){int t;long long m, n, ans, i, count;scanf("%d", &t);while (t--){scanf("%I64d%I64d", &m, &n);if (n % m) puts("0");///注意特判else{n /= m;ans = 1;for (i = 2; i * i <= n; i += 2)///不用求素数,因为范围很小(注意n在不断减小){if (n % i == 0){count = 0;while (n % i == 0){n /= i;++count;}ans *= 6 * count;}if (i == 2)--i;///小技巧}if (n > 1) ans *= 6;printf("%I64d\n", ans);}}return 0;}
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