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 PYTHON3):

Solution:
def main():
    import sys
    input=sys.stdin.readline
    mod=998244353
    N=10**5+3
    fac=[1]*(N+1)
    for i in range(1,N+1):
        fac[i]=fac[i-1]*i%mod
    inv_fac=[1]*(N+1)
    inv_fac[N]=pow(fac[N],mod-2,mod)
    for i in range(N-1,0,-1):
        inv_fac[i]=inv_fac[i+1]*(i+1)%mod
    D=int(input())
    A=[]
    for i in range(2,int(D**.5)+1):
        c=0
        while D%i==0:
            D//=i
            c+=1
        if c!=0:
        	A.append(i)
    if D>=2:
        A.append(D)
    l=len(A)
    q=int(input())
    for _ in range(q):
        u,v=map(int,input().split())
        l1=[0]*l
        l2=[0]*l
        l3=[0]*l
        for i in range(l):
            while u%A[i]==0:
                l1[i]+=1
                u//=A[i]
            while v%A[i]==0:
                l2[i]+=1
                v//=A[i]
            l3[i]=l1[i]-l2[i]
        ans1=1
        ans2=1
        s1=0
        s2=0
        for i in range(l):
            if l3[i]>=0:
                ans1=ans1*inv_fac[l3[i]]%mod
                s1+=l3[i]
            else:
                ans2=ans2*inv_fac[-l3[i]]%mod
                s2-=l3[i]
        ans1=ans1*fac[s1]%mod
        ans2=ans2*fac[s2]%mod
        print(ans1*ans2%mod)
if __name__ == '__main__':
    main()