Division Game

Problem Description

There are $k$ piles of stones in a circle, numbered from $0$ to $k - 1$, where the number of the stones in each pile is $n$ initially. You can do some round operations, where the initial round is numbered as the $1$-st round. The operation of the $i$-th round is to modify the pile of stones numbered $(i - 1) \bmod k$. In each round, you should remove from this pile some stones (at least one stone), satisfying that the number of stones in this pile before this operation is a multiple of the number of stones in this pile after operation, which means that you ought to remain at least one stone in this pile. The game is ended if there exists at least one pile containing only one stone. Given two positive integers $n$ and $k$, your task is to calculate for each pile the number of the possible operation plans that it is the last operated pile before the game is ended. The integer $n$ may be very large, so the prime-factor decomposition of $n$ will be given, in other words, if $n = \prod_{i = 1}^{m}{p_i^{e_i}}$, then the integers $m$ and $(p_i, e_i)$ $(1 \leq i \leq m)$ will be given, but the integer $n$ will not. The answer may be very large, so you only need to give the value of the answer modulo $985661441$.

Input

The input contains multiple test cases. For each test case: The first line contains two positive integers $m$ and $k$, satisfying that $1 \leq m, k \leq 10$. In next $m$ lines, the $i$-th line contains two positive integers $p_i$ and $e_i$, satisfying that $2 \leq p_i \leq 10^9,$ $e_i \geq 1,$ $\sum_{i = 1}^{m}{e_i} \leq 10^5$. It is guaranteed that $p_1, p_2, \cdots, p_m$ are distinct. About $200$ test cases in total, where no more than $5$ cases satisfy $\sum_{i = 1}^{m}{e_i} \geq 10^4$.

Output

For each test case, output " Case #$x$: $y_0$ $y_1$ $\cdots$ $y_{k - 1}$ " in one line (without quotes), where $x$ indicates the case number starting from $1$ and $y_i$ $(0 \leq i < k)$ denotes the number of the possible operation plans modulo $985661441$ for the pile numbered $i$ of corresponding case.

Sample Input

1 1

2 2

2 1

3 1

5 1

1 2

2 3

2 2

2 4

5 4

Sample Output

Case #1: 2

Case #2: 3

Case #3: 6 4

Case #4: 1499980 1281085

Source

2017 Multi-University Training Contest - Team 1

https://acm.hdu.edu.cn/showproblem.php?pid=6036