HDU Math Problems

1576

const int mod = 9973;

n = a - a / mod * mod;

a / b = ans;

ans * b = a = a / mod * mod + n;

n = b * ans - a / mod * mod;

n = b * ans + mod * y;

extended_gcd(b, mod, ans, y);

 #define PRON "hdu1576"
 #define LL "%lld"
 #include <cstdio>
 #include <cstring>
 #include <iostream>
 #include <algorithm>
 using namespace std;
 typedef long long ll;

 const int MOD = ;

 int Tcase;

 ll extended_gcd(ll a, ll b, ll & x, ll & y){
     if (b == ){
         x = , y = ;
         return a;
     }

     ll d = extended_gcd(b, a % b, x, y);
     ll temp = x;
     x = y;
     y = temp - a / b * y;

     return d;
 }

 int main(){
 #ifndef ONLINE_JUDGE
     freopen(PRON ".in", "r", stdin);
 #endif

     ll a, b, x, y;

     scanf("%d", &Tcase);
     while (Tcase --){
         scanf(LL LL, &a, &b);
         extended_gcd(b, MOD, x, y);
         x = ((x * a % MOD) + MOD) % MOD;
         printf(LL "\n", x);
     }
 }

hdu1576

2824

Σphi[i]

 #define PRON "hdu2824"
 #include <cstdio>
 #include <cstring>
 #include <iostream>
 #include <algorithm>
 using namespace std;
 typedef long long ll;

 const int MAXN = ;

 int n, m;
 ll phi[MAXN + ];

 void get_phi(){
     memset(phi, , sizeof phi);
     phi[] = ;
     for (int i = ; i <= MAXN; i ++)
         if (!phi[i]){
             for (int j = i; j <= MAXN; j += i){
                 if (!phi[j])
                     phi[j] = j;
                 phi[j] = phi[j] / i * (i - );
             }
         }
 }

 int main(){
 #ifndef ONLINE_JUDGE
     freopen(PRON ".in", "r", stdin);
 #endif

     get_phi();
     for (int i = ; i <= MAXN; i ++)
         phi[i] += phi[i - ];

     while (scanf("%d %d", &n, &m) == )
         cout << phi[m] - phi[n - ] << endl;
 }

hdu2824

1573

中国剩余定理的一般形式

 #define PRON "hdu1573"
 #include <cstdio>
 #include <cstring>
 #include <iostream>
 #include <algorithm>
 using namespace std;
 typedef int ll;

 const int MAXN =  + ;

 int Tcase, _max, n, a[MAXN], b[MAXN];

 ll extended_gcd(ll a, ll b, ll & x, ll & y){
     if (b == ){
         x = , y = ;
         return a;
     }

     ll d = extended_gcd(b, a % b, x, y);
     ll temp = x;
     x = y;
     y = temp - a / b * y;

     return d;
 }

 ll normal_crt(){
     ll m1, m2, r1, r2, x, y;

     //solve N = r1 (mod m1)
     //      N = r2 (mod m2)
     m1 = a[], r1 = b[];
     for (int i = ; i < n; i ++){
         m2 = a[i], r2 = b[i];

         //solve d = x * m1 + y * m2
         //(x, y) is the solution to the equation above
         //solve c = r2 - r1 = y * m2 - x * m1
         //(x0, y0) is the solution to the equation above
         //x0 = x * c / d, y0 = x * c / d
         ll d = extended_gcd(m1, m2, x, y);
         ll c = r2 - r1;
         if (c % d)
             return ;

         ll t = m2 / d;
         x = (x * c / d % t + t) % t;

         //r1 is the solution to the equaions from 1st to ith
         r1 += m1 * x;
         //m1 is the lcm of m1 to mi
         m1 *= t;
     }

     if (_max < r1)
         return ;

     //if (x0, y0) is one of the solution
     //(x0 + k * m2 / d, y0 - k * m1 / d) (k -> Z) also apply
     return (_max - r1) / m1 +  - (bool)(r1 == );
 }

 int main(){
 #ifndef ONLINE_JUDGE
     freopen(PRON ".in", "r", stdin);
 #endif

     scanf("%d", &Tcase);
     while (Tcase --){
         scanf("%d %d", &_max, &n);
         for (int i = ; i < n; i ++)
             scanf("%d", a + i);
         for (int i = ; i < n; i ++)
             scanf("%d", b + i);

         printf("%d\n", normal_crt());
     }
 }

1573

1370

中国剩余定理

 #define PRON "hdu1370"
 #include <cstdio>
 #include <cstring>
 #include <iostream>
 #include <algorithm>
 using namespace std;
 typedef int ll;

 const int MAXN = ;
 const int MOD = ;

 int Tcase, cnt, st, a[MAXN], b[MAXN];

 ll extended_gcd(ll a, ll b, ll & x, ll & y){
     if (b == ){
         x = , y = ;
         return a;
     }

     ll d = extended_gcd(b, a % b, x, y);
     ll temp = x;
     x = y;
     y = temp - a / b * y;

     return d;
 }

 ll inv(ll a, ll n){
     ll x, y;
     ll d = extended_gcd(a, n, x, y);
     return d ==  ? (x + n) % n : -;
 }

 ll crt(int n){
     ll ret = , m = ;

     for (int i = ; i < n; i ++)
         a[i] %= b[i], m *= b[i]; 

     for (int i = ; i < n; i ++)
         ret = (ret + a[i] * (m / b[i]) * inv(m / b[i], b[i])) % m;

     ret -= st;
     return ret + MOD * (bool)(ret <= );
 }

 int main(){
 #ifndef ONLINE_JUDGE
     freopen(PRON ".in", "r", stdin);
 #endif

     cnt = ;
     b[] = , b[] = , b[] = ;

     scanf("%d", &Tcase);
     while (scanf("%d %d %d %d", &a[], &a[], &a[], &st) ==  && !(a[] == - && a[] == - && a[] == -))
         printf("Case %d: the next triple peak occurs in %d days.\n", ++ cnt, crt());
 }

1370