A. Array Divisibility
constructive algorithms; math
An array of integers \(a_1,a_2,\cdots,a_n\) is beautiful subject to an integer \(k\) if it satisfies the following: The sum of \(a_{j}\) over all \(j\) such that \(j\) is a multiple of \(k\) and \(1 \le j \le n \), itself, is a multiple of \(k\). More formally, if \(\sum_{k | j} a_{j}\) is divisible by \(k\) for all \(1 \le j \le n\) then the array \(a\) is beautiful subject to \(k\). Here, the notation \({k|j}\) means \(k\) divides \(j\), that is, \(j\) is a multiple of \(k\). Given \(n\), find an array of positive nonzero integers, with each element less than or equal to \(10^5\) that is beautiful subject to all \(1 \le k \le n\).It can be shown that an answer always exists.
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 100\)). The description of the test cases follows.The first and only line of each test case contains a single integer \(n\) (\(1 \le n \le 100\)) — the size of the array.
For each test case, print the required array as described in the problem statement.
In the second test case, when \(n = 6\), for all integers \(k\) such that \(1 \le k \le 6\), let \(S\) be the set of all indices of the array that are divisible by \(k\). When \(k = 1\), \(S = \{1, 2, 3,4,5,6\}\) meaning \(a_1+a_2+a_3+a_4+a_5+a_6=242\) must be divisible by \(1\). When \(k = 2\), \(S = \{2,4,6\}\) meaning \(a_2+a_4+a_6=92\) must be divisible by \(2\). When \(k = 3\), \(S = \{3,6\}\) meaning \(a_3+a_6=69\) must divisible by \(3\). When \(k = 4\), \(S = \{4\}\) meaning \(a_4=32\) must divisible by \(4\). When \(k = 5\), \(S = \{5\}\) meaning \(a_5=125\) must divisible by \(5\). When \(k = 6\), \(S = \{6\}\) meaning \(a_6=54\) must divisible by \(6\). The array \(a = [10, 6, 15, 32, 125, 54]\) satisfies all of the above conditions. Hence, \(a\) is a valid array.