Divisor Analysis

#include <bits/stdc++.h>
using namespace std;

const int MAX =  1e5+5;
const int64_t MOD = 1e9+7;

int64_t phi(int64_t n) {
    int64_t result = n;
    for (int64_t i = 2; i * i <= n; i++) {
        if (n % i == 0) {
            while (n % i == 0)
                n /= i;
            result -= result / i;
        }
    }
    if (n > 1)
        result -= result / n;
    return result;
}

const int64_t MOD2 = phi(MOD);

int64_t extendedGcd(int64_t a, int64_t b, int64_t &x, int64_t &y) {
  if (b == 0) {
    x = 1; y = 0;
    return a;
  }
  int64_t g = extendedGcd(b, a%b, y, x);
  y -= x*(a/b);
  return g;
}

int64_t modularInverse(int64_t a, int64_t _MOD = MOD) {
  int64_t x, y;
  extendedGcd(a, _MOD, x, y);
  return (x%_MOD + _MOD)%_MOD;
}

int64_t modMul(int64_t a, int64_t b, int64_t _MOD = MOD) {
  a %= _MOD; b %= _MOD;
  return (__int128)a*b%_MOD;
}

int64_t binpow(int64_t base, int64_t exp) {
  base %= MOD;
  int64_t res = 1;
  while (exp > 0) {
    if (exp & 1) res = modMul(res, base);
    base = modMul(base, base);
    exp >>= 1;
  }
  return res;
}

int n;
int64_t x[MAX], k[MAX];

int main() {
  cin.tie(0)->sync_with_stdio(0);

  // Input
  cin >> n;
  for (int i = 0; i < n; ++i) {
    cin >> x[i] >> k[i];
  }

  // Number of divisors
  int64_t numOfDivisors = 1;
  for (int i = 0; i < n; ++i) {
    numOfDivisors = modMul(numOfDivisors, (k[i] + 1));
  }

  // Sum of divisors
  int64_t sumOfDivisors = 1;
  for (int i = 0; i < n; ++i) {
    sumOfDivisors = modMul(sumOfDivisors, modMul(binpow(x[i], k[i]+1)-1, modularInverse(x[i]-1)));
  }

  bool square = true;
  // Calculate N
  int64_t N = 1;
  for (int i = 0; i < n; ++i) {
    square &= (k[i] % 2 == 0);
  }

  if (square) {
    for (int i = 0; i < n; ++i) {
      N = modMul(N, binpow(x[i], k[i]/2));
    }
  } else {
    for (int i = 0; i < n; ++i) {
      N = modMul(N, binpow(x[i], k[i]));
    }
  }

  bool ok = false;
  // Number of divisors by phi(mod)
  int64_t numOfDivisorsPhiMod = 1;
  for (int i = 0; i < n; ++i) {
    if (!ok and (k[i] & 1)) {
      numOfDivisorsPhiMod = modMul(numOfDivisorsPhiMod, (k[i] + 1)/2, MOD2);
      ok = true;
    } else {
      numOfDivisorsPhiMod = modMul(numOfDivisorsPhiMod, (k[i] + 1), MOD2);
    }
  }

  // Product of divisors
  int64_t productOfDivisors = 1;

  productOfDivisors = binpow(N, numOfDivisorsPhiMod);

  cout << numOfDivisors << ' ';
  cout << sumOfDivisors << ' ';
  cout << productOfDivisors << '\n';
}