C. Almost All Multiples
greedy; number theory
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.
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 \(2 \cdot 10^5\).
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 \(-1\).
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.