复习一下莫比乌斯反演
首先很显然用一下容斥把它转化成求 \(ans=\sum_{i=1}^a \sum_{j=1}^b [{gcd(i,j)=d}]\)
我们可以定义 f(d) 和 F(d) 如下:
\(f(d)=\sum_{i=1}^N\sum_{j=1}^M[gcd(i,j)=d]\)
\(F(d)=\sum_{i=1}^N\sum_{j=1}^M[d|gcd(i,j)]\)
发现
\(\sum_{n|d}f(d)=F(n)=\lfloor\frac Nn\rfloor\lfloor\frac Mn\rfloor\)
莫比乌斯反演,得到:
\(f(n)=\sum_{n|d}\mu(\lfloor\frac dn\rfloor)F(d)\)
于是
\(ans=f(d)=\sum_{d|p}\mu(\lfloor\frac pd\rfloor)F(p)\)
换个元
\(ans=\sum_{p'}\mu(p')F( p' d ) =\sum_{p'=1}^{min(\lfloor\frac Nd\rfloor,\lfloor\frac Md\rfloor)}\mu(p')\lfloor\frac N{p'd}\rfloor \lfloor \frac M{p'd}\rfloor\)
把 \(p'\) 写作 \(p\) ,得到
\(ans=\sum_{p}\mu(p)F( p d ) =\sum_{p=1}^{min(\lfloor\frac Nd\rfloor,\lfloor\frac Md\rfloor)}\mu(p)\lfloor\frac N{pd}\rfloor \lfloor \frac M{pd}\rfloor\)
#include
using namespace std;
#define int long long
const int N = 1000005;
const int MAXN = 1000005;
bool isNotPrime[MAXN + 1];
int mu[MAXN + 1], phi[MAXN + 1], primes[MAXN + 1], cnt;
inline void euler() {
isNotPrime[0] = isNotPrime[1] = true;
mu[1] = 1;
phi[1] = 1;
for (int i = 2; i <= MAXN; i++) {
if (!isNotPrime[i]) {
primes[++cnt] = i;
mu[i] = -1;
phi[i] = i - 1;
}
for (int j = 1; j <= cnt; j++) {
int t = i * primes[j];
if (t > MAXN) break;
isNotPrime[t] = true;
if (i % primes[j] == 0) {
mu[t] = 0;
phi[t] = phi[i] * primes[j];
break;
} else {
mu[t] = -mu[i];
phi[t] = phi[i] * (primes[j] - 1);
}
}
}
for(int i=1;i<=MAXN;i++) mu[i]+=mu[i-1];
}
int solve(int n,int m,int k) {
n/=k; m/=k;
if(n==0 || m==0) return 0;
int ans=0,l=1,r=0;
if(n>m) swap(n,m);
while(l<=n) {
r=min(n/(n/l),m/(m/l));
ans+=(mu[r]-mu[l-1])*(n/l)*(m/l);
l=r+1;
}
return ans;
}
signed main() {
euler();
int t,a,b,c,d,k;
ios::sync_with_stdio(false);
cin>>t;
while(t--) {
cin>>a>>b>>c>>d>>k; --a; --c;
cout<