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 <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 = int(v);
    }
    _m_int(uint64_t v) {
      if (v >= MOD) v %= MOD;
      val = int(v);
    }
    _m_int(int v) : _m_int(int64_t(v)) {}
    _m_int(unsigned v) : _m_int(uint64_t(v)) {}
    explicit operator int() const { return val; }
    explicit operator unsigned() const { return val; }
    explicit operator int64_t() const { return val; }
    explicit operator uint64_t() const { return val; }
    explicit operator double() const { return val; }
    explicit operator long double() 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 unsigned(x % m);
      unsigned x_high = unsigned(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; }
    friend bool operator==(const _m_int &a, const _m_int &b) {
      return a.val == b.val;
    }
    friend bool operator!=(const _m_int &a, const _m_int &b) {
      return a.val != b.val;
    }
    friend bool operator<(const _m_int &a, const _m_int &b) {
      return a.val < b.val;
    }
    friend bool operator>(const _m_int &a, const _m_int &b) {
      return a.val > b.val;
    }
    friend bool operator<=(const _m_int &a, const _m_int &b) {
      return a.val <= b.val;
    }
    friend bool operator>=(const _m_int &a, const _m_int &b) {
      return a.val >= b.val;
    }
    static const int SAVE_INV = int(1e6) + 5;
    static _m_int save_inv[SAVE_INV];
    static void prepare_inv() {
      for (int64_t p = 2; p * p <= MOD; p += p % 2 + 1) assert(MOD % p != 0);
      save_inv[0] = 0;
      save_inv[1] = 1;
      for (int i = 2; i < SAVE_INV; i++)
        save_inv[i] = save_inv[MOD % i] * (MOD - MOD / i);
    }
    _m_int inv() const {
      if (save_inv[1] == 0) prepare_inv();
      if (val < SAVE_INV) return save_inv[val];
      _m_int product = 1;
      int v = val;
      while (v >= SAVE_INV) {
        product *= MOD - MOD / v;
        v = MOD % v;
      }
      return product * save_inv[v];
    }
    _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;
        p >>= 1;
        if (p > 0) a *= a;
      }
      return result;
    }
    friend ostream &operator<<(ostream &os, const _m_int &m) {
      return os << m.val;
    }
  };
  template <const int &MOD>
  _m_int<MOD> _m_int<MOD>::save_inv[_m_int<MOD>::SAVE_INV];
  extern const int MOD = 998244353;
  using mod_int = _m_int<MOD>;
  int main() {
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(0);
    int n, m;
    cin >> n >> m;
    vector<int> last_prime(m + 1, -1);
    for (int i = 2; i <= m; i++) {
      if (last_prime[i] != -1) {
        continue;
      }
      for (int j = i; j <= m; j += i) {
        last_prime[j] = i;
      }
    }
    vector<int> mobius(m + 1);
    mobius[1] = 1;
    for (int i = 2; i <= m; i++) {
      int x = i;
      bool flag = false;
      int cnt = 0;
      while (x > 1) {
        int pr = last_prime[x];
        int cur = 0;
        while (x % pr == 0) {
          x /= pr;
          cur++;
        }
        if (cur > 1) {
          flag = true;
          break;
        }
        cnt++;
      }
      if (flag) {
        continue;
      }
      mobius[i] = (cnt % 2 == 1 ? -1 : 1);
    }
    vector<int> l(n), r(n);
    for (int i = 0; i < n; i++) {
      cin >> l[i] >> r[i];
    }
    mod_int ans = 0;
    for (int g = 1; g <= m; g++) {
      if (mobius[g] == 0) {
        continue;
      }
      int s = m / g + 1;
      vector<mod_int> dp(s, 0);
      dp[0] = 1;
      for (int i = 0; i < n; i++) {
        int nl = (l[i] + g - 1) / g;
        int nr = r[i] / g;
        vector<mod_int> psum(s + 1, 0);
        for (int x = 0; x < s; x++) {
          psum[x + 1] = psum[x] + dp[x];
        }
        vector<mod_int> ndp(s, 0);
        if (nl <= nr) {
          for (int x = 0; x < s; x++) {
            ndp[x] = psum[max(x - nl + 1, 0)] - psum[max(x - nr, 0)];
          }
        }
        ndp.swap(dp);
      }
      mod_int res = 0;
      for (int x = 0; x < s; x++) {
        res += dp[x];
      }
      ans += res * mobius[g];
    }
    cout << ans << '\n';
    return 0;
  }
  