B. XOR Sequences

You are given two distinct non-negative integers x𝑥 and y𝑦. Consider two infinite sequences a1,a2,a3,…𝑎1,𝑎2,𝑎3,… and b1,b2,b3,…𝑏1,𝑏2,𝑏3,…, wherean=n⊕x𝑎𝑛=𝑛⊕𝑥;bn=n⊕y𝑏𝑛=𝑛⊕𝑦.Here, x⊕y𝑥⊕𝑦 denotes the bitwise XOR operation of integers x𝑥 and y𝑦.For example, with x=6𝑥=6, the first 88 elements of sequence a𝑎 will look as follows: [7,4,5,2,3,0,1,14,…][7,4,5,2,3,0,1,14,…]. Note that the indices of elements start with 11.Your task is to find the length of the longest common subsegment†† of sequences a𝑎 and b𝑏. In other words, find the maximum integer m𝑚 such that ai=bj,ai+1=bj+1,…,ai+m−1=bj+m−1𝑎𝑖=𝑏𝑗,𝑎𝑖+1=𝑏𝑗+1,…,𝑎𝑖+𝑚−1=𝑏𝑗+𝑚−1 for some i,j≥1𝑖,𝑗≥1.††A subsegment of sequence p𝑝 is a sequence pl,pl+1,…,pr𝑝𝑙,𝑝𝑙+1,…,𝑝𝑟, where 1≤l≤r1≤𝑙≤𝑟.InputEach test consists of multiple test cases. The first line contains a single integer t𝑡 (1≤t≤1041≤𝑡≤104) — the number of test cases. The description of the test cases follows.The only line of each test case contains two integers x𝑥 and y𝑦 (0≤x,y≤109,x≠y0≤𝑥,𝑦≤109,𝑥≠𝑦) — the parameters of the sequences.OutputFor each test case, output a single integer — the length of the longest common subsegment.ExampleinputCopy40 112 457 37316560849 14570961outputCopy18433554432NoteIn the first test case, the first 77 elements of sequences a𝑎 and b𝑏 are as follows:a=[1,2,3,4,5,6,7,…]𝑎=[1,2,3,4,5,6,7,…]b=[0,3,2,5,4,7,6,…]𝑏=[0,3,2,5,4,7,6,…]It can be shown that there isn't a positive integer k𝑘 such that the sequence [k,k+1][𝑘,𝑘+1] occurs in b𝑏 as a subsegment. So the answer is 11.In the third test case, the first 2020 elements of sequences a𝑎 and b𝑏 are as follows:a=[56,59,58,61,60,63,62,49,48,51,50,53,52,55,54,41, 40, 43, 42,45,…]𝑎=[56,59,58,61,60,63,62,49,48,51,50,53,52,55,54,41, 40, 43, 42,45,…]b=[36,39,38,33,32,35,34,45,44,47,46,41, 40, 43, 42,53,52,55,54,49,…]𝑏=[36,39,38,33,32,35,34,45,44,47,46,41, 40, 43, 42,53,52,55,54,49,…]It can be shown that one of the longest common subsegments is the subsegment [41,40,43,42][41,40,43,42] with a length of 44.

Given N, print the XOR of all numbers between (1-N).

Given an array A of size N. The value of an array is denoted by the bit-wise XOR of all elements it contains. Find the bit-wise XOR of the values of all subarrays of A.

You are given an array of integers. Find the XOR of all the pairwise sums formed by the elements of the array.Input FormatThe first line of input contains T - the number of test cases. It is followed by 2T lines, the first line contains N - the size of the array. The second line contains the elements of the array.Output FormatFor each test case, print the XOR product of all the pairwise sums of the elements from the array, separated by a newline.Constraints10 points1 <= T <= 1001 <= N <= 10000 <= A[i] <= 10540 points1 <= T <= 1001 <= N <= 1050 <= A[i] <= 105ExampleInput254 10 54 11 8615 35 25 10 15 12Output118120ExplanationTest-Case 1All the pairwise sums formed with 4 are (4 + 4), (4 + 10), (4 + 54), (4 + 11), (4 + 8) = 8, 14, 58, 15, 12All the pairwise sums formed with 10 are (10 + 4), (10 + 10), (10 + 54), (10 + 11), (10 + 8) = 14, 20, 64, 21, 18All the pairwise sums formed with 54 are (54 + 4), (54 + 10), (54 + 54), (54 + 11), (54 + 8) = 58, 64, 108, 65, 62All the pairwise sums formed with 11 are (11 + 4), (11 + 10), (11 + 54), (11 + 11), (11 + 8) = 15, 21, 65, 22, 19All the pairwise sums formed with 8 are (8 + 4), (8 + 10), (8 + 54), (8 + 11), (8 + 8) = 12, 18, 62, 19, 16XOR of all the above sums = (8 ^ 14 ^ 58 ^ 15 ^ 12) ^ (14 ^ 20 ^ 64 ^ 21 ^ 18) ^ (58 ^ 64 ^ 108 ^ 65 ^ 62) ^ (15 ^ 21 ^ 65 ^ 22 ^ 19) ^ (12 ^ 18 ^ 62 ^ 19 ^ 16) = 118

You are given a list of binary numbers [110011, 111010, 11011, 1110, 11000, 101010, 11110]. </br>You can freely choose any 2 numbers from this list. You have to choose them so that the XOR of those 2 numbers is minimized. What is the minimum XOR value possible?