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 namespace std;
  namespace modular {
  template <typename>
  struct is_modular : std::false_type {};
  template <int M>
  struct static_mint {
    static_assert(0 < M, "Module must be positive");
    using mint = static_mint;
    int val;
    static_mint() : val() {}
    static_mint(long long x) : val(x % M) {
      if (val < 0) val += M;
    }
    mint pow(long long n) const {
      mint ans = 1, x(*this);
      while (n) {
        if (n & 1) ans *= x;
        x *= x;
        n /= 2;
      }
      return ans;
    }
    mint inv() const { return pow(M - 2); }
    friend mint pow(const mint& m, long long n) { return m.pow(n); }
    friend mint inv(const mint& m) { return m.inv(); }
    mint operator+() const {
      mint m;
      m.val = val;
      return m;
    }
    mint operator-() const {
      mint m;
      m.val = M - val;
      return m;
    }
    mint& operator+=(const mint& m) {
      if ((val += m.val) >= M) val -= M;
      return *this;
    }
    mint& operator-=(const mint& m) {
      if ((val -= m.val) < 0) val += M;
      return *this;
    }
    mint& operator*=(const mint& m) {
      val = (long long)val * m.val % M;
      return *this;
    }
    mint& operator/=(const mint& m) {
      val = (long long)val * m.inv().val % M;
      return *this;
    }
    friend mint operator+(const mint& lhs, const mint& rhs) {
      return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
      return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
      return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
      return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
      return lhs.val == rhs.val;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
      return lhs.val != rhs.val;
    }
    mint& operator++() { return *this += 1; }
    mint& operator--() { return *this -= 1; }
    mint operator++(int) {
      mint result(*this);
      *this += 1;
      return result;
    }
    mint operator--(int) {
      mint result(*this);
      *this -= 1;
      return result;
    }
    template <typename T>
    explicit operator T() const {
      return T(val);
    }
    friend std::ostream& operator<<(std::ostream& os, const mint& m) {
      return os << m.val;
    }
    friend std::istream& operator>>(std::istream& is, mint& m) {
      long long x;
      is >> x;
      m = x;
      return is;
    }
  };
  template <int M>
  struct is_modular<static_mint<M>> : std::true_type {};
  template <int& _M>
  struct dynamic_mint {
    static constexpr int& M = _M;
    using mint = dynamic_mint;
    int val;
    dynamic_mint() : val() {}
    dynamic_mint(long long x) : val(x % M) {
      if (val < 0) {
        val += M;
      }
    }
    mint pow(long long n) const {
      mint ans = 1, x(*this);
      while (n) {
        if (n & 1) ans *= x;
        x *= x;
        n /= 2;
      }
      return ans;
    }
    mint inv() const { return pow(M - 2); }
    friend mint pow(const mint& m, long long n) { return m.pow(n); }
    friend mint inv(const mint& m) { return m.inv(); }
    mint operator+() const {
      mint m;
      m.val = val;
      return m;
    }
    mint operator-() const {
      mint m;
      m.val = M - val;
      return m;
    }
    mint& operator+=(const mint& m) {
      if ((val += m.val) >= M) val -= M;
      return *this;
    }
    mint& operator-=(const mint& m) {
      if ((val -= m.val) < 0) val += M;
      return *this;
    }
    mint& operator*=(const mint& m) {
      val = (long long)val * m.val % M;
      return *this;
    }
    mint& operator/=(const mint& m) {
      val = (long long)val * m.inv().val % M;
      return *this;
    }
    friend mint operator+(const mint& lhs, const mint& rhs) {
      return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
      return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
      return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
      return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
      return lhs.val == rhs.val;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
      return lhs.val != rhs.val;
    }
    mint& operator++() { return *this += 1; }
    mint& operator--() { return *this -= 1; }
    mint operator++(int) {
      mint result(*this);
      *this += 1;
      return result;
    }
    mint operator--(int) {
      mint result(*this);
      *this -= 1;
      return result;
    }
    template <typename T>
    explicit operator T() const {
      return T(val);
    }
    friend std::ostream& operator<<(std::ostream& os, const mint& m) {
      return os << m.val;
    }
    friend std::istream& operator>>(std::istream& is, mint& m) {
      long long x;
      is >> x;
      m = x;
      return is;
    }
  };
  }  // namespace modular
  using im = modular::static_mint<998244353>;
  im p[2][100001];
  im get(int i, int l, int r) {
    if (r < 0) return im(0);
    l = max(l, 0);
    return p[i][r] - (l == 0 ? 0 : p[i][l - 1]);
  }
  void solution() {
    int n, m;
    cin >> n >> m;
    vector<im> f(m + 1);
    vector<int> l(n), r(n);
    for (int i = 0; i < n; i++) {
      cin >> l[i] >> r[i];
    }
    for (int d = 1; d <= m; d++) {
      vector<int> L(n), R(n);
      bool glitch = false;
      for (int i = 0; i < n; i++) {
        L[i] = (l[i] + d - 1) / d;
        R[i] = r[i] / d;
        if (R[i] < L[i]) glitch = true;
      }
      if (glitch) continue;
      int bound = m / d;
      for (int i = 0; i <= 1; i++)
        for (int j = 0; j <= bound; j++) p[i][j] = 0;
      p[0][0] = 1;
      for (int i = 1; i <= bound; i++) p[0][i] = 1;
      int cur = 0, nxt = 1;
      for (int i = 0; i < n; i++) {
        for (int j = 0; j <= bound; j++)
          p[nxt][j] += get(cur, j - R[i], j - L[i]);
        for (int j = 1; j <= bound; j++) p[nxt][j] += p[nxt][j - 1];
        for (int j = 0; j <= bound; j++) p[cur][j] = 0;
        swap(cur, nxt);
      }
      f[d] = p[cur][bound];
    }
    for (int i = m; i >= 1; i--) {
      for (int j = i + i; j <= m; j += i) f[i] -= f[j];
    }
    cout << f[1];
  }
  signed main() {
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    int tc = 1;
    while (tc--) {
      solution();
      cout << "\n";
    }
  }
  