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;
  template <typename T>
  T mod_inv_in_range(T a, T m) {
    T x = a, y = m;
    T vx = 1, vy = 0;
    while (x) {
      T k = y / x;
      y %= x;
      vy -= k * vx;
      std::swap(x, y);
      std::swap(vx, vy);
    }
    assert(y == 1);
    return vy < 0 ? m + vy : vy;
  }
  template <typename T>
  T mod_inv(T a, T m) {
    a %= m;
    a = a < 0 ? a + m : a;
    return mod_inv_in_range(a, m);
  }
  template <int MOD_>
  struct modnum {
    static constexpr int MOD = MOD_;
    static_assert(MOD_ > 0, "MOD must be positive");
  
   private:
    using ll = long long;
    int v;
  
   public:
    modnum() : v(0) {}
    modnum(ll v_) : v(int(v_ % MOD)) {
      if (v < 0) v += MOD;
    }
    explicit operator int() const { return v; }
    friend std::ostream& operator<<(std::ostream& out, const modnum& n) {
      return out << int(n);
    }
    friend std::istream& operator>>(std::istream& in, modnum& n) {
      ll v_;
      in >> v_;
      n = modnum(v_);
      return in;
    }
    friend bool operator==(const modnum& a, const modnum& b) {
      return a.v == b.v;
    }
    friend bool operator!=(const modnum& a, const modnum& b) {
      return a.v != b.v;
    }
    modnum inv() const {
      modnum res;
      res.v = mod_inv_in_range(v, MOD);
      return res;
    }
    friend modnum inv(const modnum& m) { return m.inv(); }
    modnum neg() const {
      modnum res;
      res.v = v ? MOD - v : 0;
      return res;
    }
    friend modnum neg(const modnum& m) { return m.neg(); }
    modnum operator-() const { return neg(); }
    modnum operator+() const { return modnum(*this); }
    modnum& operator++() {
      v++;
      if (v == MOD) v = 0;
      return *this;
    }
    modnum& operator--() {
      if (v == 0) v = MOD;
      v--;
      return *this;
    }
    modnum& operator+=(const modnum& o) {
      v -= MOD - o.v;
      v = (v < 0) ? v + MOD : v;
      return *this;
    }
    modnum& operator-=(const modnum& o) {
      v -= o.v;
      v = (v < 0) ? v + MOD : v;
      return *this;
    }
    modnum& operator*=(const modnum& o) {
      v = int(ll(v) * ll(o.v) % MOD);
      return *this;
    }
    modnum& operator/=(const modnum& o) { return *this *= o.inv(); }
    friend modnum operator++(modnum& a, int) {
      modnum r = a;
      ++a;
      return r;
    }
    friend modnum operator--(modnum& a, int) {
      modnum r = a;
      --a;
      return r;
    }
    friend modnum operator+(const modnum& a, const modnum& b) {
      return modnum(a) += b;
    }
    friend modnum operator-(const modnum& a, const modnum& b) {
      return modnum(a) -= b;
    }
    friend modnum operator*(const modnum& a, const modnum& b) {
      return modnum(a) *= b;
    }
    friend modnum operator/(const modnum& a, const modnum& b) {
      return modnum(a) /= b;
    }
  };
  template <typename T>
  T pow(T a, long long b) {
    assert(b >= 0);
    T r = 1;
    while (b) {
      if (b & 1) r *= a;
      b >>= 1;
      a *= a;
    }
    return r;
  }
  const int MOD = 998244353;
  using mint = modnum<MOD>;
  const int N = 1e5 + 5;
  int n, m, l[N], r[N];
  mint f[N], dp[N], pre[N];
  int main() {
    cin.tie(0)->sync_with_stdio(false);
    cin >> n >> m;
    for (int i = 1; i <= n; ++i) cin >> l[i] >> r[i];
    for (int g = m; g >= 1; --g) {
      int T = m / g;
      for (int i = 0; i <= T; ++i) dp[i] = 0;
      dp[0] = 1;
      for (int i = 1; i <= n; ++i) {
        pre[0] = dp[0];
        for (int j = 1; j <= T; ++j) {
          pre[j] = (pre[j - 1] + dp[j]);
        }
        for (int i = 0; i <= T; ++i) dp[i] = 0;
        int lef = (l[i] + g - 1) / g;
        int rig = r[i] / g;
        if (lef <= rig) {
          for (int j = 1; j <= T; ++j) {
            int y = j - lef, x = max(j - rig, 0);
            if (x <= y) {
              dp[j] = pre[y] - (x == 0 ? 0 : pre[x - 1]);
            }
          }
        }
      }
      for (int i = 1; i <= T; ++i) {
        f[g] = (f[g] + dp[i]);
      }
      for (int k = g * 2; k <= m; k += g) {
        f[g] = f[g] - f[k];
      }
    }
    cout << f[1] << '\n';
  }
  