E. Divisor Pathstime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou 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:   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 \frac{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.InputThe first line contains a single integer D (1 \le D \le 10^{15}) — the number the graph is built from.The second line contains a single integer q (1 \le q \le 3 \cdot 10^5) — the number of queries.Each of the next q lines contains two integers v and u (1 \le v, u \le D). It is guaranteed that D is divisible by both v and u (both v and u are divisors of D).OutputPrint q integers — for each query output the number of the shortest paths between the two given vertices modulo 998244353.ExamplesInput
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
NoteIn 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). 
12
3
4 4
12 1
3 4

1 3 1 
['combinatorics', 'graphs', 'greedy', 'math', 'number theory', '*2200']