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>
  #pragma GCC optimize("O3")
  #pragma GCC target("avx2")
  #pragma GCC optimize("unroll-loops")
  using namespace std;
  template <typename T>
  ostream &operator<<(ostream &os, const vector<T> &v) {
    os << '{';
    string sep;
    for (const auto &x : v) os << sep << x, sep = ", ";
    return os << '}';
  }
  template <typename T, size_t size>
  ostream &operator<<(ostream &os, const array<T, size> &arr) {
    os << '{';
    string sep;
    for (const auto &x : arr) os << sep << x, sep = ", ";
    return os << '}';
  }
  template <typename A, typename B>
  ostream &operator<<(ostream &os, const pair<A, B> &p) {
    return os << '(' << p.first << ", " << p.second << ')';
  }
  void dbg_out() { cout << endl; }
  template <typename Head, typename... Tail>
  void dbg_out(Head H, Tail... T) {
    cout << ' ' << H;
    dbg_out(T...);
  }
  const int N = 55, mod = 998244353, bit = 61;
  const int M = 1e5 + 5;
  mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
  int getRand(int l, int r) {
    uniform_int_distribution<int> uid(l, r);
    return uid(rng);
  }
  template <const int &MOD>
  struct _m_int {
    int val;
    _m_int(int64_t v = 0) {
      if (v < 0) v = v % MOD + MOD;
      if (v >= MOD) v %= MOD;
      val = v;
    }
    static int mod_inv(int a, int m = MOD) {
      int g = m, r = a, x = 0, y = 1;
      while (r != 0) {
        int q = g / r;
        g %= r;
        swap(g, r);
        x -= q * y;
        swap(x, y);
      }
      return x < 0 ? x + m : x;
    }
    explicit operator int() const { return val; }
    explicit operator int64_t() const { return val; }
    _m_int &operator+=(const _m_int &other) {
      val -= MOD - other.val;
      if (val < 0) val += MOD;
      return *this;
    }
    _m_int &operator-=(const _m_int &other) {
      val -= other.val;
      if (val < 0) val += MOD;
      return *this;
    }
    static unsigned fast_mod(uint64_t x, unsigned m = MOD) {
      return x % m;
      unsigned x_high = x >> 32, x_low = (unsigned)x;
      unsigned quot, rem;
      asm("divl %4\n" : "=a"(quot), "=d"(rem) : "d"(x_high), "a"(x_low), "r"(m));
      return rem;
    }
    _m_int &operator*=(const _m_int &other) {
      val = fast_mod((uint64_t)val * other.val);
      return *this;
    }
    _m_int &operator/=(const _m_int &other) { return *this *= other.inv(); }
    friend _m_int operator+(const _m_int &a, const _m_int &b) {
      return _m_int(a) += b;
    }
    friend _m_int operator-(const _m_int &a, const _m_int &b) {
      return _m_int(a) -= b;
    }
    friend _m_int operator*(const _m_int &a, const _m_int &b) {
      return _m_int(a) *= b;
    }
    friend _m_int operator/(const _m_int &a, const _m_int &b) {
      return _m_int(a) /= b;
    }
    _m_int &operator++() {
      val = val == MOD - 1 ? 0 : val + 1;
      return *this;
    }
    _m_int &operator--() {
      val = val == 0 ? MOD - 1 : val - 1;
      return *this;
    }
    _m_int operator++(int) {
      _m_int before = *this;
      ++*this;
      return before;
    }
    _m_int operator--(int) {
      _m_int before = *this;
      --*this;
      return before;
    }
    _m_int operator-() const { return val == 0 ? 0 : MOD - val; }
    bool operator==(const _m_int &other) const { return val == other.val; }
    bool operator!=(const _m_int &other) const { return val != other.val; }
    bool operator<(const _m_int &other) const { return val < other.val; }
    bool operator<=(const _m_int &other) const { return val <= other.val; }
    bool operator>(const _m_int &other) const { return val > other.val; }
    bool operator>=(const _m_int &other) const { return val >= other.val; }
    _m_int inv() const { return mod_inv(val); }
    _m_int pow(int64_t p) const {
      if (p < 0) return inv().pow(-p);
      _m_int a = *this, result = 1;
      while (p > 0) {
        if (p & 1) {
          result *= a;
        }
        a *= a;
        p >>= 1;
      }
      return result;
    }
    friend ostream &operator<<(ostream &os, const _m_int &m) {
      return os << m.val;
    }
  };
  extern const int MOD = 998244353;
  using mod_int = _m_int<MOD>;
  int n, m;
  int l[N], r[N];
  mod_int dp[M];
  mod_int sum[N][M], prefSum[N][M];
  signed main() {
    ios_base::sync_with_stdio(false), cin.tie(NULL), cout.tie(NULL);
    int t = 1;
    for (long long tc = 1; tc <= t; tc++) {
      cin >> n >> m;
      for (long long i = 1; i <= n; i++) {
        cin >> l[i] >> r[i];
      }
      sum[0][0] = 1;
      for (long long i = 0; i <= M - 1; i++) {
        prefSum[0][i] = 1;
      }
      for (long long g = M - 1; g >= 1; g--) {
        int maxSum = m / g;
        for (long long i = 1; i <= n; i++) {
          for (long long j = 0; j <= maxSum; j++) {
            sum[i][j] = 0;
          }
          int currL = (l[i] + g - 1) / g;
          int currR = (r[i]) / g;
          for (long long j = currL; j <= currR; j++) {
            int oldMax = j - currL;
            sum[i][j] = prefSum[i - 1][oldMax];
          }
          for (long long j = 1 + currR; j <= maxSum; j++) {
            int oldMax = j - currL;
            int oldMin = j - currR;
            sum[i][j] = prefSum[i - 1][oldMax] - prefSum[i - 1][oldMin - 1];
          }
          prefSum[i][0] = sum[i][0];
          for (long long j = 1; j <= maxSum; j++) {
            prefSum[i][j] = sum[i][j] + prefSum[i][j - 1];
          }
        }
        dp[g] = prefSum[n][maxSum];
        for (int i = 2 * g; i < M; i += g) {
          dp[g] -= dp[i];
        }
      }
      cout << dp[1] << endl;
    }
    return 0;
  }
  