[Spoj] DCEPC13D - The Ultimate Riddle

Author            : Dipu Kumar Mohanto 
                    CSE, Batch - 6
                    BRUR.
Problem Statement : DCEPC13D - The Ultimate Riddle
Source            : Spoj
Category          : Number Theory, Combinatorics
Algorithm         : Chinese Remainder Theorem (Coprime), Lucas Theorem
Verdict           : Accepted

#include "bits/stdc++.h"
 
using namespace std;
 
#define FI              freopen("in.txt", "r", stdin)
#define FO              freopen("out.txt", "w", stdout)
#define FAST            ios_base::sync_with_stdio(false), cin.tie(NULL), cout.tie(NULL)
 
#define FOR(i, n)       for (int i = 1; i <= n; i++)
#define For(i, n)       for (int i = 0; i < n; i++)
#define ROF(i, n)       for (int i = n; i >= 1; i--)
#define Rof(i, n)       for (int i = n-1; i >= 0; i--)
#define FORI(i, n)      for (auto i : n)
 
#define ll              long long
#define ull             unsigned long long
#define vi              vector <int>
#define vl              vector <ll>
#define pii             pair <int, int>
#define pll             pair <ll, ll>
#define mk              make_pair
#define ff              first
#define ss              second
#define eb              emplace_back
#define em              emplace
#define pb              push_back
#define ppb             pop_back
#define All(a)          a.begin(), a.end()
#define memo(a, b)      memset(a, b, sizeof a)
#define Sort(a)         sort(All(a))
#define ED(a)           Sort(a), a.erase(unique(All(a)), a.end())
#define rev(a)          reverse(All(a))
#define sz(a)           (int)a.size()
#define max3(a, b, c)   max(a, max(b, c))
#define min3(a, b, c)   min(a, min(b, c))
#define maxAll(a)       *max_element(All(a))
#define minAll(a)       *min_element(All(a))
#define allUpper(a)     transform(All(a), a.begin(), :: toupper)
#define allLower(a)     transform(All(a), a.begin(), :: tolower)
#define endl            '\n'
#define nl              puts("")
#define ub              upper_bound
#define lb              lower_bound
#define Exp             exp(1.0)
#define PIE             2*acos(0.0)
#define Sin(a)          sin(((a)*PIE)/180.0)
#define EPS             1e-9
 
// Important Functions
// mo compare function : friend bool operator < (mo p, mo q) { int pb = p.l / block; int qb = q.l / block; if (pb != qb) return p.l < q.l; return (p.r < q.r) ^ (p.l / block % 2); }
// factorial MOD inverse : ll inv[maxn]; inline ll factorialInverse() { inv[1] = 1; for (int i = 2; i < maxn; i++) inv[i] = (MOD – (MOD / i)) * inv[MOD % i) % MOD; }
 
/**
  * uses of PBDS:
  * 1) *X.find_by_order(k) : returns k'th element
  * 2) X.order_of_key(k)   : count total element less than k
  * 3) All other operations same as set
**/
#include "ext/pb_ds/assoc_container.hpp"
#include "ext/pb_ds/tree_policy.hpp"
using namespace __gnu_pbds;
 
template <typename T> using orderset = tree <T, null_type, less <T>, rb_tree_tag, tree_order_statistics_node_update>;
 
/**
  * uses of rope data structure:
  * Initialize                 : rope <data_type> Rope;
  * Insert single element      : Rope.push_back()
  * Insert a block of elements : Rope.insert(position after insert, newRope)
  * Erase a block of elements  : Rope.erase(starting position of deletion, size of deletion)
  * Other Operations           : Same as std::vector
  * Iterator                   : Rope.mutable_begin(), Rope.mutable_end(), Rope.mutable_rbegin(), Rope.mutable_rend()
  * Caution                    : Never use -- Rope.begin() iterator
**/
#include "ext/rope"
using namespace __gnu_cxx;
 
// rope <int> Rope;
 
// int dr[] = {1, -1, 0, 0}; // 4 Direction
// int dc[] = {0, 0, 1, -1};
// int dr[] = {0, 0, 1, -1, 1, 1, -1, -1}; // 8 Direction
// int dc[] = {1, -1, 0, 0, 1, -1, 1, -1};
// int dr[] = {-1, 1, -2, -2, -1, 1, 2, 2}; // knight Moves
// int dc[] = {-2, -2, -1, 1, 2, 2, 1, -1};
 
#define trace1(x)                           cerr << #x << ": " << x << endl;
#define trace2(x, y)                        cerr << #x << ": " << x << " | " << #y << ": " << y << endl;
#define trace3(x, y, z)                     cerr << #x << ": " << x << " | " << #y << ": " << y << " | " << #z << ": " << z << endl;
#define trace4(a, b, c, d)                  cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": " << c << " | " << #d << ": " << d << endl;
#define trace5(a, b, c, d, e)               cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": " << c << " | " << #d << ": " << d << " | " << #e << ": " << e << endl;
#define trace6(a, b, c, d, e, f)            cerr << #a << ": " << a << " | " << #b << ": " << b << " | " << #c << ": " << c << " | " << #d << ": " << d << " | " << #e << ": " << e << " | " << #f << ": " << f << endl;
 
inline int setbit(int mask, int pos)        { return mask |= (1 << pos); }
inline int resetbit(int mask, int pos)      { return mask &= ~(1 << pos); }
inline int togglebit(int mask, int pos)     { return mask ^= (1 << pos); }
inline bool checkbit(int mask, int pos)     { return (bool)(mask & (1 << pos)); }
 
#define popcount(mask)                       __builtin_popcount(mask) // count set bit
#define popcountLL(mask)                     __builtin_popcountll(mask) // for long long
 
inline int read()                           { int a; scanf("%d", &a); return a; }
inline ll readLL()                          { ll a; scanf("%lld", &a); return a; }
inline double readDD()                      { double a; scanf("%lf", &a); return a; }
 
template <typename T> string toString(T num) { stringstream ss; ss << num; return ss.str(); }
int toInt(string s)                          { int num; istringstream iss(s); iss >> num; return num;  }
ll toLLong(string s)                         { ll num; istringstream iss(s); iss >> num; return num; }
 
#define inf             1e17
#define mod             1000000007
 
static const int maxn = 1e6 + 5;
static const int logn = 18;
 
inline void extendedEuclidAlgorithm(ll a, ll b, ll &x, ll &y, ll &d)
{
      if (b == 0)
      {
            x = 1;
            y = 0;
            d = a;
            return;
      }
      extendedEuclidAlgorithm(b, a%b, x, y, d);
      ll x1 = y;
      ll y1 = x - (a / b) * y;
      x = x1;
      y = y1;
}
 
inline pll chineseRemainderTheorem(vl A, vl M)
{
      if (sz(A) != sz(M)) return pll(-1, -1);
      int n = sz(A);
      ll a1 = A[0];
      ll m1 = M[0];
      for (int i = 1; i < n; i++)
      {
            ll a2 = A[i];
            ll m2 = M[i];
            ll p, q, d;
            extendedEuclidAlgorithm(m1, m2, p, q, d);
            ll x = (a1 * m2 * q + a2 * m1 * p) % (m1 * m2);
            a1 = x;
            m1 = m1 * m2;
      }
      if (a1 < 0) a1 += m1;
      return pll(a1, m1);
}
 
inline ll bigMod(ll a, ll p, ll m)
{
      if (p == 0) return 1 % m;
      if (p == 1) return a % m;
      if (p & 1) return (a % m * bigMod(a, p-1, m) % m) % m;
      ll ret = bigMod(a, p>>1, m);
      return (ret % m * ret % m) % m;
}
 
inline ll modInverse(ll a, ll m)
{
      return bigMod(a, m-2, m);
}
 
ll fact[maxn];
 
inline ll nCr(ll n, ll r, ll p)
{
      if (n < r) return 0;
      ll upor = fact[n] % p;
      ll nich = (fact[r] * fact[n-r]) % p;
      nich = modInverse(nich, p);
      return (upor * nich) % p;
}
 
ll lucas(ll n, ll r, ll p)
{
      if (n == 0 and r == 0) return 1;
      ll ni = n % p;
      ll ri = r % p;
      return (nCr(ni, ri, p) * lucas(n/p, r/p, p)) % p;
}
 
inline ll get_nCr(ll n, ll r, ll p)
{
      fact[0] = 1;
      for (int i = 1; i < p; i++) fact[i] = (i * fact[i-1]) % p;
      return lucas(n, r, p);
}
 
inline vl primeFactor(ll num)
{
      vl pf;
      for (int i = 2; i < 50; i++) if (num % i == 0) pf.eb(i), num /= i;
      return pf;
}
 
int main()
{
      //FI;
      int tc = read();
      FOR(tcase, tc)
      {
            ll N = readLL();
            ll R = readLL();
            ll P = readLL();
            vl pf = primeFactor(P);
            vl mf;
            for (ll p : pf)
            {
                  ll m = get_nCr(N, R, p);
                  mf.eb(m);
            }
            pll crt = chineseRemainderTheorem(mf, pf);
            printf("%lld\n", crt.ff);
      }
}