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:
import io, os
import sys
input = io.BytesIO(os.read(0,os.fstat(0).st_size)).readline

D=int(input())
q=int(input())
mod=998244353
Q=[tuple(map(int,input().split())) for i in range(q)]

from math import gcd
from math import sqrt

L=[]
Q2=[]
for x,y in Q:
    GCD=gcd(x,y)
    x//=GCD
    y//=GCD
    Q2.append((x,y))
    L.append(x)
    L.append(y)

L.sort()
FACTOR=set()

for l in L:
    for f in FACTOR:
        while l%f==0:
            l//=f
    if l==1:
        continue
    else:
        L=int(sqrt(l))
        for i in range(2,L+2):
            while l%i==0:
                FACTOR.add(i)
                l//=i
        if l!=1:
            FACTOR.add(l)


FACT=[1]
for i in range(1,2*10**5+1):
    FACT.append(FACT[-1]*i%mod)

FACT_INV=[pow(FACT[-1],mod-2,mod)]
for i in range(2*10**5,0,-1):
    FACT_INV.append(FACT_INV[-1]*i%mod)

FACT_INV.reverse()

def Combi(a,b):
    if 0<=b<=a:
        return FACT[a]*FACT_INV[b]*FACT_INV[a-b]%mod
    else:
        return 0


def FACTORIZE(x):

    F=[]

    for f in FACTOR:
        #print(f,x)
        a=0
        while x%f==0:
            a+=1
            x//=f
        if a!=0:
            F.append(a)
    return F

for x,y in Q2:
    ANS=1
    
    F=FACTORIZE(x)
    S=sum(F)
    for f in F:
        ANS=ANS*Combi(S,f)%mod
        S-=f

    F=FACTORIZE(y)
    S=sum(F)
    for f in F:
        ANS=ANS*Combi(S,f)%mod
        S-=f

    sys.stdout.write(str(ANS%mod)+"\n")