1334E · Divisor Paths

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 $D=12$:

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 $998244353$.

Input Format: The 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 $D$).

Output Format: Print $q$ integers — for each query output the number of the shortest paths between the two given vertices modulo $998244353$.

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 $1+1+1=3$).