Problem: You are given a positive integer D. Let's build the following graph from it: 

  * each vertex is a divisor of D (not necessarily prime, 1 and D itself are also included); 
  * two vertices x and y (x > y) have an undirected edge between them if x is divisible by y and \frac x y is a prime; 
  * the weight of an edge is the number of divisors of x that are not divisors of y. 



For example, here is the graph for D=12: 

<image>

Edge (4,12) has weight 3 because 12 has divisors [1,2,3,4,6,12] and 4 has divisors [1,2,4]. Thus, there are 3 divisors of 12 that are not divisors of 4 — [3,6,12].

There is no edge between 3 and 2 because 3 is not divisible by 2. There is no edge between 12 and 3 because 12/3=4 is not a prime.

Let the length of the path between some vertices v and u in the graph be the total weight of edges on it. For example, path [(1, 2), (2, 6), (6, 12), (12, 4), (4, 2), (2, 6)] has length 1+2+2+3+1+2=11. The empty path has length 0.

So the shortest path between two vertices v and u is the path that has the minimal possible length.

Two paths a and b are different if there is either a different number of edges in them or there is a position i such that a_i and b_i are different edges.

You are given q queries of the following form: 

  * v u — calculate the number of the shortest paths between vertices v and u. 



The answer for each query might be large so print it modulo 998244353.

Input

The first line contains a single integer D (1 ≤ D ≤ 10^{15}) — the number the graph is built from.

The second line contains a single integer q (1 ≤ q ≤ 3 ⋅ 10^5) — the number of queries.

Each of the next q lines contains two integers v and u (1 ≤ v, u ≤ D). It is guaranteed that D is divisible by both v and u (both v and u are divisors of D).

Output

Print q integers — for each query output the number of the shortest paths between the two given vertices modulo 998244353.

Examples

Input


12
3
4 4
12 1
3 4


Output


1
3
1


Input


1
1
1 1


Output


1


Input


288807105787200
4
46 482955026400
12556830686400 897
414 12556830686400
4443186242880 325


Output


547558588
277147129
457421435
702277623

Note

In the first example: 

  * The first query is only the empty path — length 0; 
  * The second query are paths [(12, 4), (4, 2), (2, 1)] (length 3+1+1=5), [(12, 6), (6, 2), (2, 1)] (length 2+2+1=5) and [(12, 6), (6, 3), (3, 1)] (length 2+2+1=5). 
  * The third query is only the path [(3, 1), (1, 2), (2, 4)] (length 1+1+1=3). 

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
const int N = 100005;
const int inf = 1000 * 1000 * 1000;
const int mod = 998244353;
mt19937 myrand(chrono::steady_clock::now().time_since_epoch().count());
int c[105][105];
long long d;
int q;
vector<long long> pr;
long long gcd(long long a, long long b) {
  if (b == 0) return a;
  return gcd(b, a % b);
}
int cnt(long long a, long long b) {
  if (a < b) swap(a, b);
  a /= b;
  int s = 0;
  vector<int> mas;
  for (int i = 0; i < (int)pr.size(); i++) {
    int c = 0;
    while (a % pr[i] == 0) {
      a /= pr[i];
      ++c;
    }
    if (c != 0) mas.push_back(c);
    s += c;
  }
  int answ = 1;
  for (int i = 0; i < (int)mas.size(); i++) {
    answ = answ * 1ll * c[s][mas[i]] % mod;
    s -= mas[i];
  }
  return answ;
}
int main() {
  c[0][0] = 1;
  for (int i = 1; i <= 100; i++) {
    c[i][0] = 1;
    for (int j = 1; j <= i; j++) {
      c[i][j] = (c[i - 1][j] + c[i - 1][j - 1]) % mod;
    }
  }
  cin >> d;
  scanf("%d", &q);
  if (d == 1) {
    while (q--) {
      long long u, v;
      scanf("%lld%lld", &u, &v);
      printf("1\n");
    }
    return 0;
  }
  for (long long i = 2; i * i <= d; i++) {
    if (d % i == 0) {
      pr.push_back(i);
      while (d % i == 0) {
        d /= i;
      }
    }
  }
  if (d != 1) {
    pr.push_back(d);
  }
  while (q--) {
    long long u, v;
    scanf("%lld%lld", &u, &v);
    long long p = gcd(u, v);
    printf("%d\n", (cnt(u, p) * 1ll * cnt(v, p)) % mod);
  }
  return 0;
}