GCD
题目连接:
http://acm.hdu.edu.cn/showproblem.php?pid=1695
Description
Given 5 integers: a, b, c, d, k, you're to find x in a...b, y in c...d that GCD(x, y) = k. GCD(x, y) means the greatest common divisor of x and y. Since the number of choices may be very large, you're only required to output the total number of different number pairs.
Please notice that, (x=5, y=7) and (x=7, y=5) are considered to be the same.
Yoiu can assume that a = c = 1 in all test cases.
Input
T
a b c d k
Output
ans
Sample Input
1
1 3 1 5 1
Sample Output
9
Hint
题意
问你gcd(i,j)=k有多少对,其中b>=i>=a,d>=j>=c
其中a和c恒等于1
题解:
很显然,我们知道一个结论,gcd(i,j)=k,b>=i>=a,d>=j>=c这个恒等于
gcd(i,j)=1,b/k>=i>=1,d/k>=j>=1
然后怎么办呢?我们暴力枚举每一个1<=i<=b/k的数,看在1<=j<=d/k里面有多少个和他互质的就好了
这个我们可以用容斥来做。
代码
#include<bits/stdc++.h>
using namespace std;
const int maxn = 2e5+5;
vector<int>pri[maxn];
void pre()
{
for(int i=2;i<100007;i++)
{
int now = i;
for(int j=2;j*j<=now;j++)
{
if(now%j==0)
{
pri[i].push_back(j);
while(now%j==0)
now/=j;
}
if(now==1)break;
}
if(now>1)
pri[i].push_back(now);
}
}
int solve(int x,int tot)
{
int res = 0;
for(int i=1;i<(1<<pri[x].size());i++)
{
int num = 0;
int tmp = 1;
for(int j=0;j<pri[x].size();j++)
{
if((i>>j)&1)
{
num++;
tmp*=pri[x][j];
}
}
if(num%2==1)res+=tot/tmp;
else res-=tot/tmp;
}
return tot-res;
}
int main()
{
pre();
int t;
scanf("%d",&t);
for(int cas=1;cas<=t;cas++)
{
int a,b,c,d,k;
scanf("%d%d%d%d%d",&a,&b,&c,&d,&k);
if(k==0)
{
printf("Case %d: 0
",cas);
continue;
}
b/=k,d/=k;
if(b<d)swap(b,d);
long long ans = 0;
for(int i=1;i<=b;i++)
ans+=solve(i,min(i,d));
printf("Case %d: %lld
",cas,ans);
}
}