中国剩余定理
Chivas-Regal
#
# HDUOJ1370_Biorhythms
# 🔗
https://acm.dingbacode.com/showproblem.php?pid=1370
# 💡
问题简化成下列三个式子:
m[1]=23, m[2]=28, m[3]=33
x≡a[1](mod m[1])
x≡a[2](mod m[2])
x≡a[3](mod m[3])
解此中国剩余定理即可
最后求得的Ex_crt()-d要改成最小正整数
# ✅
/*
________ _ ________ _
/ ______| | | | __ | | |
/ / | | | |__| | | |
| | | |___ _ _ _ ___ _ _____ | ___| ______ _____ ___ _ | |
| | | __ \ |_| | | | | | _\| | | ____| | |\ \ | __ | | _ | | _\| | | |
| | | | \ | _ | | | | | | \ | | \___ | | \ \ | |_/ _| | |_| | | | \ | | |
\ \______ | | | | | | \ |_| / | |_/ | ___/ | | | \ \ | /_ \__ | | |_/ | | |
Author : \________| |_| |_| |_| \___/ |___/|_| |_____| _________|__| \__\ |______| | | |___/|_| |_|
____| |
\_____/
*/
#include <unordered_map>
#include <algorithm>
#include <iostream>
#include <cstring>
#include <utility>
#include <string>
#include <vector>
#include <cstdio>
#include <stack>
#include <queue>
#include <cmath>
#include <map>
#include <set>
#define G 10.0
#define LNF 1e18
#define EPS 1e-6
#define PI acos(-1.0)
#define INF 0x7FFFFFFF
#define ll long long
#define ull unsigned long long
#define LOWBIT(x) ((x) & (-x))
#define LOWBD(a, x) lower_bound(a.begin(), a.end(), x) - a.begin()
#define UPPBD(a, x) upper_bound(a.begin(), a.end(), x) - a.begin()
#define TEST(a) cout << "---------" << a << "---------" << '\n'
#define CHIVAS_ int main()
#define _REGAL exit(0)
#define SP system("pause")
#define IOS ios::sync_with_stdio(false)
//#define map unordered_map
#define _int(a) int a; cin >> a
#define _ll(a) ll a; cin >> a
#define _char(a) char a; cin >> a
#define _string(a) string a; cin >> a
#define _vectorInt(a, n) vector<int>a(n); cin >> a
#define _vectorLL(a, b) vector<ll>a(n); cin >> a
#define PB(x) push_back(x)
#define ALL(a) a.begin(),a.end()
#define MEM(a, b) memset(a, b, sizeof(a))
#define EACH_CASE(cass) for (cass = inputInt(); cass; cass--)
#define LS l, mid, rt << 1
#define RS mid + 1, r, rt << 1 | 1
#define GETMID (l + r) >> 1
using namespace std;
template<typename T> inline T MAX(T a, T b){return a > b? a : b;}
template<typename T> inline T MIN(T a, T b){return a > b? b : a;}
template<typename T> inline void SWAP(T &a, T &b){T tp = a; a = b; b = tp;}
template<typename T> inline T GCD(T a, T b){return b > 0? GCD(b, a % b) : a;}
template<typename T> inline void ADD_TO_VEC_int(T &n, vector<T> &vec){vec.clear(); cin >> n; for(int i = 0; i < n; i ++){T x; cin >> x, vec.PB(x);}}
template<typename T> inline pair<T, T> MaxInVector_ll(vector<T> vec){T MaxVal = -LNF, MaxId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MaxVal < vec[i]) MaxVal = vec[i], MaxId = i; return {MaxVal, MaxId};}
template<typename T> inline pair<T, T> MinInVector_ll(vector<T> vec){T MinVal = LNF, MinId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MinVal > vec[i]) MinVal = vec[i], MinId = i; return {MinVal, MinId};}
template<typename T> inline pair<T, T> MaxInVector_int(vector<T> vec){T MaxVal = -INF, MaxId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MaxVal < vec[i]) MaxVal = vec[i], MaxId = i; return {MaxVal, MaxId};}
template<typename T> inline pair<T, T> MinInVector_int(vector<T> vec){T MinVal = INF, MinId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MinVal > vec[i]) MinVal = vec[i], MinId = i; return {MinVal, MinId};}
template<typename T> inline pair<map<T, T>, vector<T> > DIV(T n){T nn = n;map<T, T> cnt;vector<T> div;for(ll i = 2; i * i <= nn; i ++){while(n % i == 0){if(!cnt[i]) div.push_back(i);cnt[i] ++;n /= i;}}if(n != 1){if(!cnt[n]) div.push_back(n);cnt[n] ++;n /= n;}return {cnt, div};}
template<typename T> vector<T>& operator-- (vector<T> &v){for (auto& i : v) --i; return v;}
template<typename T> vector<T>& operator++ (vector<T> &v){for (auto& i : v) ++i; return v;}
template<typename T> istream& operator>>(istream& is, vector<T> &v){for (auto& i : v) is >> i; return is;}
template<typename T> ostream& operator<<(ostream& os, vector<T> v){for (auto& i : v) os << i << ' '; return os;}
inline int inputInt(){int X=0; bool flag=1; char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') flag=0; ch=getchar();}while(ch>='0'&&ch<='9') {X=(X<<1)+(X<<3)+ch-'0'; ch=getchar();}if(flag) return X;return ~(X-1);}
inline void outInt(int X){if(X<0) {putchar('-'); X=~(X-1);}int s[20],top=0;while(X) {s[++top]=X%10; X/=10;}if(!top) s[++top]=0;while(top) putchar(s[top--]+'0');}
inline ll inputLL(){ll X=0; bool flag=1; char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') flag=0; ch=getchar();}while(ch>='0'&&ch<='9') {X=(X<<1)+(X<<3)+ch-'0'; ch=getchar();}if(flag) return X;return ~(X-1); }
inline void outLL(ll X){if(X<0) {putchar('-'); X=~(X-1);}ll s[20],top=0;while(X) {s[++top]=X%10; X/=10;}if(!top) s[++top]=0;while(top) putchar(s[top--]+'0');}
const int N = 3;
int m[N + 1] = {0, 23, 28, 33};
int a[N + 1];
int cass, d;
inline int Ex_gcd(int a, int b, int &x, int &y){
if (b == 0) { x = 1, y = 0; return a; }
int d = Ex_gcd(b, a % b, x, y);
int tmp = x; x = y; y = tmp - a / b * y;
return d;
}
inline int ksc(int a, int b, int mod){
int res = 0;
while(b){
if (b & 1) res = (res + a) % mod;
a = a * 2 % mod;
b >>= 1;
}return res;
}
inline int Ex_crt(){
int X = a[1], M = m[1];
for(int i = 2, t, y; i <= N; i ++){
int gcd = Ex_gcd(M, m[i], t, y), mDIVgcd = m[i] / gcd;
int c = (a[i] - X % m[i] + m[i]) % m[i];
if(c % gcd) return -1;
t = ksc(t, c / gcd, mDIVgcd);
X += t * M;
M *= mDIVgcd;
X = (X % M + M) % M;
}return X;
}
CHIVAS_{
EACH_CASE(cass){
int CasId = 0;
while(scanf("%d%d%d%d", &a[1], &a[2], &a[3], &d) == 4 && a[1] + a[2] + a[3] + d != -4){
printf("Case %d: the next triple peak occurs in ", ++ CasId);
if(Ex_crt() == d) outInt(21252);
else outInt(((Ex_crt() - d) % 21252 + 21252) % 21252);
puts(" days.");
}
}
_REGAL;
};
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# HDUOJ3579_HelloKiki
# 🔗
https://acm.dingbacode.com/showproblem.php?pid=3579
# 💡
很裸的模板题
转化成: x≡a[i](mod m[i])
Ex_crt求解即可
# ✅
/*
________ _ ________ _
/ ______| | | | __ | | |
/ / | | | |__| | | |
| | | |___ _ _ _ ___ _ _____ | ___| ______ _____ ___ _ | |
| | | __ \ |_| | | | | | _\| | | ____| | |\ \ | __ | | _ | | _\| | | |
| | | | \ | _ | | | | | | \ | | \___ | | \ \ | |_/ _| | |_| | | | \ | | |
\ \______ | | | | | | \ |_| / | |_/ | ___/ | | | \ \ | /_ \__ | | |_/ | | |
Author : \________| |_| |_| |_| \___/ |___/|_| |_____| _________|__| \__\ |______| | | |___/|_| |_|
____| |
\_____/
*/
#include <unordered_map>
#include <algorithm>
#include <iostream>
#include <cstring>
#include <utility>
#include <string>
#include <vector>
#include <cstdio>
#include <stack>
#include <queue>
#include <cmath>
#include <map>
#include <set>
#define G 10.0
#define LNF 1e18
#define EPS 1e-6
#define PI acos(-1.0)
#define INF 0x7FFFFFFF
#define ll long long
#define ull unsigned long long
#define LOWBIT(x) ((x) & (-x))
#define LOWBD(a, x) lower_bound(a.begin(), a.end(), x) - a.begin()
#define UPPBD(a, x) upper_bound(a.begin(), a.end(), x) - a.begin()
#define TEST(a) cout << "---------" << a << "---------" << '\n'
#define CHIVAS_ int main()
#define _REGAL exit(0)
#define SP system("pause")
#define IOS ios::sync_with_stdio(false)
//#define map unordered_map
#define _int(a) int a; cin >> a
#define _ll(a) ll a; cin >> a
#define _char(a) char a; cin >> a
#define _string(a) string a; cin >> a
#define _vectorInt(a, n) vector<int>a(n); cin >> a
#define _vectorLL(a, b) vector<ll>a(n); cin >> a
#define PB(x) push_back(x)
#define ALL(a) a.begin(),a.end()
#define MEM(a, b) memset(a, b, sizeof(a))
#define EACH_CASE(cass) for (cass = inputInt(); cass; cass--)
#define LS l, mid, rt << 1
#define RS mid + 1, r, rt << 1 | 1
#define GETMID (l + r) >> 1
using namespace std;
template<typename T> inline T MAX(T a, T b){return a > b? a : b;}
template<typename T> inline T MIN(T a, T b){return a > b? b : a;}
template<typename T> inline void SWAP(T &a, T &b){T tp = a; a = b; b = tp;}
template<typename T> inline T GCD(T a, T b){return b > 0? GCD(b, a % b) : a;}
template<typename T> inline void ADD_TO_VEC_int(T &n, vector<T> &vec){vec.clear(); cin >> n; for(int i = 0; i < n; i ++){T x; cin >> x, vec.PB(x);}}
template<typename T> inline pair<T, T> MaxInVector_ll(vector<T> vec){T MaxVal = -LNF, MaxId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MaxVal < vec[i]) MaxVal = vec[i], MaxId = i; return {MaxVal, MaxId};}
template<typename T> inline pair<T, T> MinInVector_ll(vector<T> vec){T MinVal = LNF, MinId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MinVal > vec[i]) MinVal = vec[i], MinId = i; return {MinVal, MinId};}
template<typename T> inline pair<T, T> MaxInVector_int(vector<T> vec){T MaxVal = -INF, MaxId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MaxVal < vec[i]) MaxVal = vec[i], MaxId = i; return {MaxVal, MaxId};}
template<typename T> inline pair<T, T> MinInVector_int(vector<T> vec){T MinVal = INF, MinId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MinVal > vec[i]) MinVal = vec[i], MinId = i; return {MinVal, MinId};}
template<typename T> inline pair<map<T, T>, vector<T> > DIV(T n){T nn = n;map<T, T> cnt;vector<T> div;for(ll i = 2; i * i <= nn; i ++){while(n % i == 0){if(!cnt[i]) div.push_back(i);cnt[i] ++;n /= i;}}if(n != 1){if(!cnt[n]) div.push_back(n);cnt[n] ++;n /= n;}return {cnt, div};}
template<typename T> vector<T>& operator-- (vector<T> &v){for (auto& i : v) --i; return v;}
template<typename T> vector<T>& operator++ (vector<T> &v){for (auto& i : v) ++i; return v;}
template<typename T> istream& operator>>(istream& is, vector<T> &v){for (auto& i : v) is >> i; return is;}
template<typename T> ostream& operator<<(ostream& os, vector<T> v){for (auto& i : v) os << i << ' '; return os;}
inline int inputInt(){int X=0; bool flag=1; char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') flag=0; ch=getchar();}while(ch>='0'&&ch<='9') {X=(X<<1)+(X<<3)+ch-'0'; ch=getchar();}if(flag) return X;return ~(X-1);}
inline void outInt(int X){if(X<0) {putchar('-'); X=~(X-1);}int s[20],top=0;while(X) {s[++top]=X%10; X/=10;}if(!top) s[++top]=0;while(top) putchar(s[top--]+'0');}
inline ll inputLL(){ll X=0; bool flag=1; char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') flag=0; ch=getchar();}while(ch>='0'&&ch<='9') {X=(X<<1)+(X<<3)+ch-'0'; ch=getchar();}if(flag) return X;return ~(X-1); }
inline void outLL(ll X){if(X<0) {putchar('-'); X=~(X-1);}ll s[20],top=0;while(X) {s[++top]=X%10; X/=10;}if(!top) s[++top]=0;while(top) putchar(s[top--]+'0');}
const int N = 10;
int m[N], a[N], n;
int cass, CasId;
inline int Ex_gcd(int a, int b, int &x, int &y){
if (b == 0) { x = 1, y = 0; return a; }
int d = Ex_gcd(b, a % b, x, y);
int tmp = x; x = y; y = tmp - a / b * y;
return d;
}
inline int ksc(int a, int b, int mod){
int res = 0;
while(b){
if (b & 1) res = (res + a) % mod;
a = a * 2 % mod;
b >>= 1;
}return res;
}
inline int Ex_crt(){
int X = a[1], M = m[1];
for(int i = 2, t, y; i <= n; i ++){
int gcd = Ex_gcd(M, m[i], t, y), mDIVgcd = m[i] / gcd;
int c = (a[i] - X % m[i] + m[i]) % m[i];
if(c % gcd) return -1;
t = ksc(t, c / gcd, mDIVgcd);
X += t * M;
M *= mDIVgcd;
X = (X % M + M) % M;
}return X == 0? M : X;
}
CHIVAS_{
EACH_CASE(cass){
n = inputInt();
for(int i = 1; i <= n; i ++) m[i] = inputInt();
for(int i = 1; i <= n; i ++) a[i] = inputInt();
printf("Case %d: %d\n", ++ CasId, Ex_crt());
}
_REGAL;
};
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# README
📕【模板】扩展中国剩余定理
# 🔗
https://www.luogu.com.cn/problem/P4777
# 💡
# ✅
/*
________ _ ________ _
/ ______| | | | __ | | |
/ / | | | |__| | | |
| | | |___ _ _ _ ___ _ _____ | ___| ______ _____ ___ _ | |
| | | __ \ |_| | | | | | _\| | | ____| | |\ \ | __ | | _ | | _\| | | |
| | | | \ | _ | | | | | | \ | | \___ | | \ \ | |_/ _| | |_| | | | \ | | |
\ \______ | | | | | | \ |_| / | |_/ | ___/ | | | \ \ | /_ \__ | | |_/ | | |
Author : \________| |_| |_| |_| \___/ |___/|_| |_____| _________|__| \__\ |______| | | |___/|_| |_|
____| |
\_____/
*/
#include <unordered_map>
#include <algorithm>
#include <iostream>
#include <cstring>
#include <utility>
#include <string>
#include <vector>
#include <cstdio>
#include <stack>
#include <queue>
#include <cmath>
#include <map>
#include <set>
#define G 10.0
#define LNF 1e18
#define EPS 1e-6
#define PI acos(-1.0)
#define INF 0x7FFFFFFF
#define ll long long
#define ull unsigned long long
#define LOWBIT(x) ((x) & (-x))
#define LOWBD(a, x) lower_bound(a.begin(), a.end(), x) - a.begin()
#define UPPBD(a, x) upper_bound(a.begin(), a.end(), x) - a.begin()
#define TEST(a) cout << "---------" << a << "---------" << '\n'
#define CHIVAS_ int main()
#define _REGAL exit(0)
#define SP system("pause")
#define IOS ios::sync_with_stdio(false)
//#define map unordered_map
#define _int(a) int a; cin >> a
#define _ll(a) ll a; cin >> a
#define _char(a) char a; cin >> a
#define _string(a) string a; cin >> a
#define _vectorInt(a, n) vector<int>a(n); cin >> a
#define _vectorLL(a, b) vector<ll>a(n); cin >> a
#define PB(x) push_back(x)
#define ALL(a) a.begin(),a.end()
#define MEM(a, b) memset(a, b, sizeof(a))
#define EACH_CASE(cass) for (cass = inputInt(); cass; cass--)
#define LS l, mid, rt << 1
#define RS mid + 1, r, rt << 1 | 1
#define GETMID (l + r) >> 1
using namespace std;
template<typename T> inline T MAX(T a, T b){return a > b? a : b;}
template<typename T> inline T MIN(T a, T b){return a > b? b : a;}
template<typename T> inline void SWAP(T &a, T &b){T tp = a; a = b; b = tp;}
template<typename T> inline T GCD(T a, T b){return b > 0? GCD(b, a % b) : a;}
template<typename T> inline void ADD_TO_VEC_int(T &n, vector<T> &vec){vec.clear(); cin >> n; for(int i = 0; i < n; i ++){T x; cin >> x, vec.PB(x);}}
template<typename T> inline pair<T, T> MaxInVector_ll(vector<T> vec){T MaxVal = -LNF, MaxId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MaxVal < vec[i]) MaxVal = vec[i], MaxId = i; return {MaxVal, MaxId};}
template<typename T> inline pair<T, T> MinInVector_ll(vector<T> vec){T MinVal = LNF, MinId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MinVal > vec[i]) MinVal = vec[i], MinId = i; return {MinVal, MinId};}
template<typename T> inline pair<T, T> MaxInVector_int(vector<T> vec){T MaxVal = -INF, MaxId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MaxVal < vec[i]) MaxVal = vec[i], MaxId = i; return {MaxVal, MaxId};}
template<typename T> inline pair<T, T> MinInVector_int(vector<T> vec){T MinVal = INF, MinId = 0;for(int i = 0; i < (int)vec.size(); i ++) if(MinVal > vec[i]) MinVal = vec[i], MinId = i; return {MinVal, MinId};}
template<typename T> inline pair<map<T, T>, vector<T> > DIV(T n){T nn = n;map<T, T> cnt;vector<T> div;for(ll i = 2; i * i <= nn; i ++){while(n % i == 0){if(!cnt[i]) div.push_back(i);cnt[i] ++;n /= i;}}if(n != 1){if(!cnt[n]) div.push_back(n);cnt[n] ++;n /= n;}return {cnt, div};}
template<typename T> vector<T>& operator-- (vector<T> &v){for (auto& i : v) --i; return v;}
template<typename T> vector<T>& operator++ (vector<T> &v){for (auto& i : v) ++i; return v;}
template<typename T> istream& operator>>(istream& is, vector<T> &v){for (auto& i : v) is >> i; return is;}
template<typename T> ostream& operator<<(ostream& os, vector<T> v){for (auto& i : v) os << i << ' '; return os;}
inline int inputInt(){int X=0; bool flag=1; char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') flag=0; ch=getchar();}while(ch>='0'&&ch<='9') {X=(X<<1)+(X<<3)+ch-'0'; ch=getchar();}if(flag) return X;return ~(X-1);}
inline void outInt(int X){if(X<0) {putchar('-'); X=~(X-1);}int s[20],top=0;while(X) {s[++top]=X%10; X/=10;}if(!top) s[++top]=0;while(top) putchar(s[top--]+'0');}
inline ll inputLL(){ll X=0; bool flag=1; char ch=getchar();while(ch<'0'||ch>'9') {if(ch=='-') flag=0; ch=getchar();}while(ch>='0'&&ch<='9') {X=(X<<1)+(X<<3)+ch-'0'; ch=getchar();}if(flag) return X;return ~(X-1); }
inline void outLL(ll X){if(X<0) {putchar('-'); X=~(X-1);}ll s[20],top=0;while(X) {s[++top]=X%10; X/=10;}if(!top) s[++top]=0;while(top) putchar(s[top--]+'0');}
const int N = 100010;
ll a[N], m[N], n; // a[i]X ≡ c(mod m[i])
inline ll Ex_gcd(ll a, ll b, ll &x, ll &y){
if(b == 0) { x = 1; y = 0; return a; }
ll d = Ex_gcd(b, a % b, x, y);
ll tmp = x; x = y; y = tmp - (a / b) * y;
return d;
}
inline ll ksc(ll a, ll b, ll mod){//快速乘、好取模
ll res = 0;
while(b){
if(b & 1) res = (res + a) % mod;
a = (a + a) % mod;
b >>= 1;
}return res;
}
inline ll Ex_crt(){
ll X = a[1], M = m[1]; //前一步的X，前一步的lcm
for (ll i = 2, t, y; i <= n; i ++){
ll gcd = Ex_gcd(M, m[i], t, y), miDIVgcd = m[i] / gcd; // 求得gcd，并使m[i]约分一下好乘进M里面
ll c = (a[i] - X % m[i] + m[i]) % m[i]; // ax≡c(mod b) 等式右侧的c，并让他变成可行的最小正整数
if(c % gcd) return -1;
t = ksc(t, c / gcd, miDIVgcd); // 因为扩欧求得的是等号右侧为gcd时的x解，而此时等号右端为c，需要让X乘上c/gcd个t，此时先给t变了再说
X += t * M;
M *= miDIVgcd; //计算LCM
X = (X % M + M) % M; //保持最小非负整数解
}return X == 0? M : X; //输出最小正整数
}
CHIVAS_{
n = inputLL();
for(int i = 1; i <= n; i ++) m[i] = inputLL(), a[i] = inputLL();
outLL(Ex_crt());
_REGAL;
};
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