description: Mocha wants to be an astrologer. There are n stars which can be seen in Zhijiang, and the brightness of the i-th star is a_i. 
  
  Mocha considers that these n stars form a constellation, and she uses (a_1,a_2,…,a_n) to show its state. A state is called mathematical if all of the following three conditions are satisfied:
  
    * For all i (1≤ i≤ n), a_i is an integer in the range [l_i, r_i].
    * ∑  _{i=1} ^ n a_i ≤ m.
    * \gcd(a_1,a_2,…,a_n)=1.
  
  
  
  Here, \gcd(a_1,a_2,…,a_n) denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers a_1,a_2,…,a_n.
  
  Mocha is wondering how many different mathematical states of this constellation exist. Because the answer may be large, you must find it modulo 998 244 353.
  
  Two states (a_1,a_2,…,a_n) and (b_1,b_2,…,b_n) are considered different if there exists i (1≤ i≤ n) such that a_i ≠ b_i.
  
  Input
  
  The first line contains two integers n and m (2 ≤ n ≤ 50, 1 ≤ m ≤ 10^5) — the number of stars and the upper bound of the sum of the brightness of stars.
  
  Each of the next n lines contains two integers l_i and r_i (1 ≤ l_i ≤ r_i ≤ m) — the range of the brightness of the i-th star.
  
  Output
  
  Print a single integer — the number of different mathematical states of this constellation, modulo 998 244 353.
  
  Examples
  
  Input
  
  
  2 4
  1 3
  1 2
  
  
  Output
  
  
  4
  
  Input
  
  
  5 10
  1 10
  1 10
  1 10
  1 10
  1 10
  
  
  Output
  
  
  251
  
  Input
  
  
  5 100
  1 94
  1 96
  1 91
  4 96
  6 97
  
  
  Output
  
  
  47464146
  
  Note
  
  In the first example, there are 4 different mathematical states of this constellation:
  
    * a_1=1, a_2=1.
    * a_1=1, a_2=2.
    * a_1=2, a_2=1.
    * a_1=3, a_2=1.
solution: #include <bits/stdc++.h>
  using ll = long long;
  const int M = 1e5 + 5;
  const int mod = 998244353;
  std::vector<std::pair<int, int> > lr;
  int mobius[M];
  int cnt[M];
  int is_composite[M];
  std::vector<int> primes;
  int solve(std::vector<int> &L, std::vector<int> &R, int m) {
    int n = L.size();
    for (int i = 0; i < n; i++) {
      m -= L[i];
      R[i] -= L[i];
    }
    if (m < 0) {
      return 0;
    }
    std::vector<ll> dp(m + 2, 0);
    dp[0] = 1;
    for (int r : R) {
      std::vector<ll> nways(m + 2, 0);
      for (int i = 0; i <= m; i++) {
        nways[i] += dp[i];
        int e = std::min(m + 1, i + r);
        nways[e] -= dp[i];
      }
      for (int i = 1; i <= m + 1; i++) {
        nways[i] = (nways[i] % mod + nways[i - 1] % mod) % mod;
      }
      std::swap(dp, nways);
    }
    int ans = 0;
    for (int i = 0; i <= m; i++) {
      ans = (ans + dp[i]) % mod;
    }
    return ans;
  }
  int main() {
    std::ios_base::sync_with_stdio(false);
    int n, m;
    std::cin >> n >> m;
    lr.resize(n);
    for (int i = 0; i < n; i++) {
      std::cin >> lr[i].first >> lr[i].second;
      lr[i].second++;
    }
    std::fill(is_composite, is_composite + m + 1, false);
    std::fill(cnt, cnt + m + 1, 0);
    is_composite[0] = true;
    is_composite[1] = true;
    mobius[0] = 1;
    mobius[1] = 1;
    for (int i = 2; i <= m; i++) {
      if (!is_composite[i]) {
        mobius[i] = -1;
        primes.push_back(i);
        cnt[i] = 1;
      }
      for (int j = 0; j < primes.size() && i * primes[j] <= m; j++) {
        is_composite[i * primes[j]] = true;
        cnt[i * primes[j]] += cnt[i];
        mobius[i * primes[j]] = mobius[primes[j]] * mobius[i];
        if (i % primes[j] == 0) {
          int powp = pow(primes[j], cnt[i]);
          mobius[i * primes[j]] = 0;
          break;
        }
      }
    }
    std::vector<int> L(n);
    std::vector<int> R(n);
    int ans = 0;
    for (int i = 1; i <= m; i++) {
      if (mobius[i] != 0) {
        for (int j = 0; j < n; j++) {
          L[j] = (lr[j].first + i - 1) / i;
          R[j] = (lr[j].second + i - 1) / i;
        }
        int tmp = solve(L, R, m / i);
        int tmp2 = mobius[i] * tmp;
        ans = (ans + tmp2) % mod;
        if (ans < 0) {
          ans += mod;
        }
      }
    }
    std::cout << ans << std::endl;
    return 0;
  }
  