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.util.*;

public class E
{
    PrintWriter out = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
    BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
    StringTokenizer tok;
    long MOD = 998244353;
    long[] fact = new long[1000];
    long[] inv = new long[10000];
    long[] invfact = new long[10000];
    long[] spn;
    ArrayList<Integer> primes = new ArrayList<>();
    HashMap<Long, Long> map = new HashMap<>();

    public void go() throws IOException
    {
        fact[0] = 1;
        for (int i = 1; i < fact.length; i++) {
            fact[i] = fact[i-1] * i % MOD;
        }
        invfact[0] = inverse(fact[0]);
        for (int i = 1; i < fact.length; i++) {
            invfact[i] = invfact[i-1] * inverse(i) % MOD;
        }
        ntok();
        long D = lpar();
        boolean[] sieve = new boolean[(int)Math.sqrt(D) + 10];
        // spn = new long[(int)Math.sqrt(D) + 10];
        Arrays.fill(sieve, true);
        for (int i = 2; i < sieve.length; i++) {
            if (sieve[i]) {
                primes.add(i);
                for (int e = 2; e*i < sieve.length; e++) {
                    // if (sieve[e*i]) {
                    //     spn[e*i] = i;
                    // }
                    sieve[e*i] = false;
                }
            }
        }
        ntok();
        int q = ipar();
        for (int qq = 0; qq < q; qq++) {
            ntok();
            long a = lpar();
            long b = lpar();
            if (a % b == 0) {
                out.println(getPaths(a, b));
            } else if (b % a == 0) {
                out.println(getPaths(b, a));
            } else {
                long c = gcd(a, b);
                out.println(getPaths(a, c) * getPaths(b, c) % MOD);
            }
        }

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

    public long getPaths(long a, long b) {
        long val = a / b;
        long val2 = val;
        if (map.containsKey(val)) {
            return map.get(val);
        }
        int total = 0;
        long div = 1;
        for (int i = 0; i < primes.size(); i++) {
            if (primes.get(i) > val) {
                break;
            }
            long p = primes.get(i);
            // out.println("DFFDD " + val);
            int c = 0;
            while (val % p == 0) {
                total++;
                c++;
                val /= p;
            }
            if (c > 0) {
                div *= invfact[c];
                div %= MOD;
            }
        }
        if (val > 1) {
            total++;
        }
        // out.println("    " + total);
        long ans = fact[total] * div % MOD;
        map.put(val2, ans);
        return ans;
    }

    public long inverse(long a) {
        return modPow(a, MOD-2);
    }

    public long modPow(long a, long b) {
        if (b == 0) {
            return 1;
        }
        if (b % 2 == 0) {
            return modPow(a*a%MOD, b/2);
        }
        return modPow(a*a%MOD, b/2) * a % MOD;
    }

    public long gcd(long a, long b) {
        if (b == 0) {
            return a;
        }
        return gcd(b, a%b);
    }

    public HashMap<Long, Integer> getDivMap(long D) {
        LinkedList<Long> divisors = new LinkedList<>();
        for (long i = (int)Math.sqrt(D); i >= 1; i++) {
            if (D % i == 0) {
                divisors.addFirst(i);
                if (D / i != i) {
                    divisors.addLast(D / i);
                }
            }
        }
        ArrayList<Long> divisors2 = new ArrayList<>(divisors);
        HashMap<Long, Integer> map = new HashMap<>();
        for (int i = 0; i < divisors2.size(); i++) {
            map.put(divisors2.get(i), i);
        }
        return map;
    }

    public void ntok() throws IOException
    {
        tok = new StringTokenizer(in.readLine());
    }

    public int ipar()
    {
        return Integer.parseInt(tok.nextToken());
    }

    public int[] iapar(int n)
    {
        int[] arr = new int[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = ipar();
        }
        return arr;
    }

    public long lpar()
    {
        return Long.parseLong(tok.nextToken());
    }

    public long[] lapar(int n)
    {
        long[] arr = new long[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = lpar();
        }
        return arr;
    }

    public double dpar()
    {
        return Double.parseDouble(tok.nextToken());
    }

    public String spar()
    {
        return tok.nextToken();
    }

    public static void main(String[] args) throws IOException
    {
        new E().go();
    }
}