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 JAVA):

Solution:
import java.util.*;
import java.io.*;

public class E85
{
    static HashMap<Long, Long> f;
    public static void main(String [] args)
    {
        MyScanner sc = new MyScanner();
        PrintWriter out = new PrintWriter(new BufferedOutputStream(System.out));
        long d = sc.nextLong(); int q = sc.nextInt();
        f = new HashMap<>();
        factorize(d);
        for (int i = 0; i < q; i++) {
            long a = sc.nextLong(); long b = sc.nextLong();
            long u = Math.min(a, b); long v = Math.max(a, b);
            ArrayList<Integer> uExp = new ArrayList<>(); ArrayList<Integer> vExp = new ArrayList<>();
            for (Long prime: f.keySet()) {
                int uCount = 0; int vCount = 0;
                while (u % prime == 0) {
                    uCount++;
                    u /= prime;
                }
                while (v % prime == 0) {
                    vCount++;
                    v /= prime;
                }
                uExp.add(uCount); vExp.add(vCount);
            }
            ArrayList<Integer> pos = new ArrayList<>(); ArrayList<Integer> neg = new ArrayList<>();
            for (int j = 0; j < uExp.size(); j++) {
                if (uExp.get(j) > vExp.get(j)) {
                    pos.add(uExp.get(j) - vExp.get(j));
                }
                if (uExp.get(j) < vExp.get(j)) {
                    neg.add(vExp.get(j) - uExp.get(j));
                }
            }
            long pSum = 0; for (int j= 0; j < pos.size(); j++) pSum += pos.get(j);
            long nSum = 0; for (int j= 0; j < neg.size(); j++) nSum += neg.get(j);
            long pWays = 1, nWays = 1; fac = new long[(int)Math.max(pSum, nSum) + 3];
            fac[0] = 1;
            for (int j = 1; j < fac.length; j++) {
                fac[j] = fac[j - 1] * j % MOD;
            }
            for (int j = 0; j < pos.size(); j++) {
                pWays = (pWays * fac[pos.get(j)]) % MOD;
            }
            for (int j = 0; j < neg.size(); j++) {
                nWays = (nWays * fac[neg.get(j)]) % MOD;
            }
            long pInv = inv((int) (pWays) % MOD);
            long nInv = inv((int) (nWays) % MOD);
            pWays = (fac[(int)pSum] * pInv) % MOD;
            nWays = (fac[(int) nSum] * nInv) % MOD;
            long ans = (pWays * nWays) % MOD;
            out.println(ans);
        }
        out.close();
    }

    static long[] fac;
    static int MOD = 998244353;

    static long nck(int n, int k) {
        if (n < k)
            return 0;
        long den = inv((int) (fac[k] * fac[n - k] % MOD));
        return fac[n] * den % MOD;
    }

    static int pow(long b, int e) {
        long ans = 1;
        while (e > 0) {
            if (e % 2 == 1)
                ans = ans * b % MOD;
            e >>= 1;
            b = b * b % MOD;
        }
        return (int) ans;
    }

    static int inv(int x) {
        return pow(x, MOD - 2);
    }

    static void factorize(long d) {
        if (d == 1) return;
        long orig = d;
        for (long i = 2; i <= Math.sqrt(orig); i++) {
            if (d == 1) return;
            while (d % i == 0) {
                f.put(i, f.getOrDefault(i, (long) 0) + 1);
                d /= i;
            }
        }
        if (d != 1) {
            f.put(d, (long)1);
        }
    }





    //-----------MyScanner class for faster input----------
    public static class MyScanner {
        BufferedReader br;
        StringTokenizer st;

        public MyScanner() {
            br = new BufferedReader(new InputStreamReader(System.in));
        }

        String next() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }

        String nextLine(){
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }



    }

}