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.io.*;
import java.math.BigInteger;
import java.util.*;

/*
Прокрастинирую
*/

public class Main {

    static FastReader in;
    static PrintWriter out;
    static Random rand = new Random();
    static final int INF = (int) (1e9 + 10), MOD = (int) (998244353), LOGN = 60;
    static final long IINF = (long) (2e18 + 10);
    static final int N = (int) (1e5 + 5);
    static long[] fact, fact1;

    static long pow(long a, long b) {
        long res = 1;
        while (b != 0) {
            if ((b & 1) != 0) {
                res = (res * a) % MOD;
            }
            a = (a * a) % MOD;
            b >>= 1;
        }
        return res;
    }

    static long inv(long a) {
        return pow(a, MOD - 2);
    }

    static long gcd(long a, long b) {
        return b == 0 ? a : gcd(b, a % b);
    }

    static class Pair {
        long a;
        int b;

        Pair(long a, int b) {
            this.a = a;
            this.b = b;
        }
    }

    static HashMap<Long, ArrayList<Pair>> memo = new HashMap<>();
    static ArrayList<Pair> factor(long a) {
        if (memo.containsKey(a)) {
            return memo.get(a);
        }
        ArrayList<Pair> res = new ArrayList<>();
        long aa = a;
        for (long i = 2; i * i <= aa; i++) {
            if (aa % i != 0) continue;
            int cnt = 0;
            while (aa % i == 0) {
                aa /= i;
                cnt++;
            }
            res.add(new Pair(i, cnt));
        }
        if (aa > 1) {
            res.add(new Pair(aa, 1));
            aa = 1;
        }
        memo.put(a, res);
        return res;
    }

    static long go(long a, long b) {
        ArrayList<Pair> f = factor(a / b);
        long res = 1;
        int total = 0;
        for (Pair p : f) {
            total += p.b;
            res = (res * fact1[p.b]) % MOD;
        }
        res = (res * fact[total]) % MOD;
        return res;
    }

    static void solve() {
        long d = in.nextLong();

        fact = new long[LOGN];
        fact1 = new long[LOGN];
        fact[0] = fact1[0] = 1;
        for (int i = 1; i < LOGN; i++) {
            fact[i] = (fact[i - 1] * i) % MOD;
            fact1[i] = inv(fact[i]);
        }

        int q = in.nextInt();
        for (int i = 0; i < q; i++) {
            long v = in.nextLong();
            long u = in.nextLong();
            long g = gcd(v, u);
            out.println(go(v, g) * go(u, g) % MOD);
        }
    }

    public static void main(String[] args) throws FileNotFoundException, InterruptedException {
        in = new FastReader(System.in);
//        in = new FastReader(new FileInputStream("input.txt"));
        out = new PrintWriter(System.out);
//        out = new PrintWriter(new FileOutputStream("output.txt"));

        Thread thread = new Thread(null, () -> {
            int tests = 1;
//            tests = in.nextInt();

            while (tests-- > 0) {
                solve();
            }
        }, "Go", 1 << 28);
        thread.start();
        thread.join();

//        out.flush();
        out.close();
    }

    static class FastReader {
        BufferedReader br;
        StringTokenizer st;

        FastReader(InputStream is) {
            br = new BufferedReader(new InputStreamReader(is));
        }

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

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

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

        String next() {
            while (st == null || !st.hasMoreTokens()) {
                st = new StringTokenizer(nextLine());
            }
            return st.nextToken();
        }

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