题目大意
给定 \(a,b,d\),求
\[\sum\limits_{i=1}^a\sum\limits_{j=1}^b[\gcd(i,j)=d] \]题目分析
令 \(a\le b\)。
显然先把 \(d\) 消掉(令 \(a'=\left\lfloor\dfrac{a}{d}\right\rfloor,b'=\left\lfloor\dfrac{b}{d}\right\rfloor\)):
\[\sum\limits_{i=1}^{a'}\sum\limits_{j=1}^{b'}[\gcd(i,j)=1] \]又因为 \(\epsilon(x)=\sum\limits_{d|x}\mu(d)\)。(\(\epsilon(x)=[x=1]\))
所以
\[\sum\limits_{i=1}^{a'}\sum\limits_{j=1}^{b'}\sum\limits_{x|\gcd(i,j)}\mu(x) \] \[=\sum\limits_{i=1}^{a'}\sum\limits_{j=1}^{b'}\sum\limits_{x=1}^{a'}\mu(x)\cdot[x|\gcd(i,j)] \] \[=\sum\limits_{x=1}^{a'}\mu(x)\left\lfloor\dfrac{a'}{x}\right\rfloor\left\lfloor\dfrac{b'}{x}\right\rfloor \] \[=\sum\limits_{x=1}^{a'}\mu(x)\left\lfloor\dfrac{a}{x\cdot d}\right\rfloor\left\lfloor\dfrac{b}{x\cdot d}\right\rfloor \]因为是多组询问,所以前面的 \(\sum\limits_{x=1}^{a'}\mu(x)\) 弄个前缀和,后面的 \(\left\lfloor\dfrac{a}{x\cdot d}\right\rfloor\left\lfloor\dfrac{b}{x\cdot d}\right\rfloor\) 整数分块即可。
代码
//2022/1/15
//2022/1/16
#define _CRT_SECURE_NO_WARNINGS
#include <iostream>
#include <cstdio>
#include <climits>//need "INT_MAX","INT_MIN"
#include <cstring>//need "memset"
#include <algorithm>
#define int long long
#define enter() putchar(10)
#define debug(c,que) cerr<<#c<<" = "<<c<<que
#define cek(c) puts(c)
#define blow(arr,st,ed,w) for(register int i=(st);i<=(ed);i++)cout<<arr[i]<<w;
#define speed_up() cin.tie(0),cout.tie(0)
#define endl "\n"
#define Input_Int(n,a) for(register int i=1;i<=n;i++)scanf("%d",a+i);
#define Input_Long(n,a) for(register long long i=1;i<=n;i++)scanf("%lld",a+i);
#define mst(a,k) memset(a,k,sizeof(a))
namespace Newstd
{
inline int read()
{
int x=0,k=1;
char ch=getchar();
while(ch<'0' || ch>'9')
{
if(ch=='-')
{
k=-1;
}
ch=getchar();
}
while(ch>='0' && ch<='9')
{
x=(x<<1)+(x<<3)+ch-'0';
ch=getchar();
}
return x*k;
}
inline void write(int x)
{
if(x<0)
{
putchar('-');
x=-x;
}
if(x>9)
{
write(x/10);
}
putchar(x%10+'0');
}
}
using namespace Newstd;
using namespace std;
const int ma=5e4+5;
int p[ma],mu[ma],sum[ma];
bool is[ma];
int T,a,b,d;
int idx;
inline void init(int R)
{
mu[1]=1,is[1]=true;
for(register int i=2;i<R;i++)
{
if(is[i]==false)
{
p[++idx]=i;
mu[i]=-1;
}
for(register int j=1;j<=idx && i*p[j]<R;j++)
{
is[i*p[j]]=true;
if(i%p[j]==0)
{
mu[i*p[j]]=0;
break;
}
mu[i*p[j]]=-mu[i];
}
}
for(register int i=1;i<R;i++)
{
sum[i]=sum[i-1]+mu[i];
}
}
#undef int
int main(void)
{
#define int long long
T=read();
init(ma);
while(T--)
{
a=read(),b=read(),d=read();
int n=a/d,m=b/d;
if(n<m)//keep n>=m
{
swap(n,m);
}
int res(0);
for(register int l=1,r;l<=m;l=r+1)
{
r=min(n/(n/l),m/(m/l));
res+=(sum[r]-sum[l-1])*((n/l)*(m/l));
}
printf("%lld\n",res);
}
return 0;
}