Almost All Multiples

1000ms

262144K

Description:

Given two integers $$$n$$$ and $$$x$$$, a permutation$$$^{\dagger}$$$ $$$p$$$ of length $$$n$$$ is called funny if $$$p_i$$$ is a multiple of $$$i$$$ for all $$$1 \leq i \leq n - 1$$$, $$$p_n = 1$$$, and $$$p_1 = x$$$.

Find the lexicographically minimal$$$^{\ddagger}$$$ funny permutation, or report that no such permutation exists.

$$$^{\dagger}$$$ A permutation of length $$$n$$$ is an array consisting of each of the integers from $$$1$$$ to $$$n$$$ exactly once.

$$$^{\ddagger}$$$ Let $$$a$$$ and $$$b$$$ be permutations of length $$$n$$$. Then $$$a$$$ is lexicographically smaller than $$$b$$$ if in the first position $$$i$$$ where $$$a$$$ and $$$b$$$ differ, $$$a_i < b_i$$$. A permutation is lexicographically minimal if it is lexicographically smaller than all other permutations.

Input:

The input consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $$$n$$$ and $$$x$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$; $$$1 < x \leq n$$$).

The sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, if the answer exists, output $$$n$$$ distinct integers $$$p_1, p_2, \dots, p_n$$$ ($$$1 \leq p_i \leq n$$$) — the lexicographically minimal funny permutation $$$p$$$. Otherwise, output $$$-1$$$.

Sample Input:

Sample Output:

3 2 1 
2 4 3 1 
-1

Note:

In the first test case, the permutation $$$[3,2,1]$$$ satisfies all the conditions: $$$p_1=3$$$, $$$p_3=1$$$, and:

$$$p_1=3$$$ is a multiple of $$$1$$$.
$$$p_2=2$$$ is a multiple of $$$2$$$.

In the second test case, the permutation $$$[2,4,3,1]$$$ satisfies all the conditions: $$$p_1=2$$$, $$$p_4=1$$$, and:

$$$p_1=2$$$ is a multiple of $$$1$$$.
$$$p_2=4$$$ is a multiple of $$$2$$$.
$$$p_3=3$$$ is a multiple of $$$3$$$.

We can show that these permutations are lexicographically minimal.

No such permutations exist in the third test case.

Informação

Provedor Codeforces

Código CF1758C