You are given a 0-indexed integer array nums, and you are allowed to traverse between its indices. You can traverse between index i and index j, i != j, if and only if gcd(nums[i], nums[j]) > 1, where gcd is the greatest common divisor.Your task is to determine if for every pair of indices i and j in nums, where i < j, there exists a sequence of traversals that can take us from i to j.Return true if it is possible to traverse between all such pairs of indices, or false otherwise. Example 1:Input: nums = [2,3,6]Output: trueExplanation: In this example, there are 3 possible pairs of indices: (0, 1), (0, 2), and (1, 2).To go from index 0 to index 1, we can use the sequence of traversals 0 -> 2 -> 1, where we move from index 0 to index 2 because gcd(nums[0], nums[2]) = gcd(2, 6) = 2 > 1, and then move from index 2 to index 1 because gcd(nums[2], nums[1]) = gcd(6, 3) = 3 > 1.To go from index 0 to index 2, we can just go directly because gcd(nums[0], nums[2]) = gcd(2, 6) = 2 > 1. Likewise, to go from index 1 to index 2, we can just go directly because gcd(nums[1], nums[2]) = gcd(3, 6) = 3 > 1.Example 2:Input: nums = [3,9,5]Output: falseExplanation: No sequence of traversals can take us from index 0 to index 2 in this example. So, we return false.Example 3:Input: nums = [4,3,12,8]Output: trueExplanation: There are 6 possible pairs of indices to traverse between: (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), and (2, 3). A valid sequence of traversals exists for each pair, so we return true.

Arun is tasked with developing a binary search algorithm to efficiently locate the index of the first occurrence of a number divisible by 3 in a given sorted array of integers. Write a program to help Arun navigate the array elements and locate the required index. If a number divisible by 3 is found, return its index; otherwise, gracefully handle the absence of such a number.Input format :The first line consists of an integer n, representing the size of the array.The second line consists of n space-separated integers representing the elements of the sorted

You are given a 0-indexed integer array nums having length n, an integer indexDifference, and an integer valueDifference.Your task is to find two indices i and j, both in the range [0, n - 1], that satisfy the following conditions:abs(i - j) >= indexDifference, andabs(nums[i] - nums[j]) >= valueDifferenceReturn an integer array answer, where answer = [i, j] if there are two such indices, and answer = [-1, -1] otherwise. If there are multiple choices for the two indices, return any of them.Note: i and j may be equal. Example 1:Input: nums = [5,1,4,1], indexDifference = 2, valueDifference = 4Output: [0,3]Explanation: In this example, i = 0 and j = 3 can be selected.abs(0 - 3) >= 2 and abs(nums[0] - nums[3]) >= 4.Hence, a valid answer is [0,3].[3,0] is also a valid answer.Example 2:Input: nums = [2,1], indexDifference = 0, valueDifference = 0Output: [0,0]Explanation: In this example, i = 0 and j = 0 can be selected.abs(0 - 0) >= 0 and abs(nums[0] - nums[0]) >= 0.Hence, a valid answer is [0,0].Other valid answers are [0,1], [1,0], and [1,1].Example 3:Input: nums = [1,2,3], indexDifference = 2, valueDifference = 4Output: [-1,-1]Explanation: In this example, it can be shown that it is impossible to find two indices that satisfy both conditions.Hence, [-1,-1] is returned. Constraints:1 <= n == nums.length <= 1050 <= nums[i] <= 1090 <= indexDifference <= 1050 <= valueDifference <= 109

You are given 2 integer arrays nums1 and nums2 of lengths n and m respectively. You are also given a positive integer k.A pair (i, j) is called good if nums1[i] is divisible by nums2[j] * k (0 <= i <= n - 1, 0 <= j <= m - 1).Return the total number of good pairs. Example 1:Input: nums1 = [1,3,4], nums2 = [1,3,4], k = 1Output: 5Explanation:The 5 good pairs are (0, 0), (1, 0), (1, 1), (2, 0), and (2, 2).Example 2:Input: nums1 = [1,2,4,12], nums2 = [2,4], k = 3Output: 2Explanation:The 2 good pairs are (3, 0) and (3, 1). Constraints:1 <= n, m <= 1051 <= nums1[i], nums2[j] <= 1061 <= k <= 103C++ 1class Solution {2public:3    long long numberOfPairs(vector<int>& nums1, vector<int>& nums2, int k) {4        5   }6};

You have been given an array 'A' of N integers. You need to find the maximum value of j - i subjected to the constraint of A[i] <= A[j], where ‘i’ and ‘j’ are the indices of the array.For example :If 'A' = {3, 5, 4, 1}then the output will be 2.Maximum value occurs for the pair (3, 4)Detailed explanation ( Input/output format, Notes, Images )Constraints:1 <= T <= 1001 <= N <= 10 ^ 4-10 ^ 5 <= A[i] <= 10 ^ 5Time limit: 1 sec.Sample Input 1:1934 8 10 3 2 80 30 33 1Sample Output 1:6Explanation:Maximum value occurs for the pair (8, 33)Sample Input 2:1109 2 3 4 5 6 7 8 18 0Sample Output 2:8Explanation:Maximum value occurs for the pair (9, 18)

For k𝑘 positive integers x1,x2,…,xk𝑥1,𝑥2,…,𝑥𝑘, the value gcd(x1,x2,…,xk)gcd(𝑥1,𝑥2,…,𝑥𝑘) is the greatest common divisor of the integers x1,x2,…,xk𝑥1,𝑥2,…,𝑥𝑘 — the largest integer z𝑧 such that all the integers x1,x2,…,xk𝑥1,𝑥2,…,𝑥𝑘 are divisible by z𝑧.You are given three arrays a1,a2,…,an𝑎1,𝑎2,…,𝑎𝑛, b1,b2,…,bn𝑏1,𝑏2,…,𝑏𝑛 and c1,c2,…,cn𝑐1,𝑐2,…,𝑐𝑛 of length n𝑛, containing positive integers.You also have a machine that allows you to swap ai𝑎𝑖 and bi𝑏𝑖 for any i𝑖 (1≤i≤n1≤𝑖≤𝑛). Each swap costs you ci𝑐𝑖 coins.Find the maximum possible value ofgcd(a1,a2,…,an)+gcd(b1,b2,…,bn)gcd(𝑎1,𝑎2,…,𝑎𝑛)+gcd(𝑏1,𝑏2,…,𝑏𝑛)that you can get by paying in total at most d𝑑 coins for swapping some elements. The amount of coins you have changes a lot, so find the answer to this question for each of the q𝑞 possible values d1,d2,…,dq𝑑1,𝑑2,…,𝑑𝑞.InputThere are two integers on the first line — the numbers n𝑛 and q𝑞 (1≤n≤5⋅1051≤𝑛≤5⋅105, 1≤q≤5⋅1051≤𝑞≤5⋅105).On the second line, there are n𝑛 integers — the numbers a1,a2,…,an𝑎1,𝑎2,…,𝑎𝑛 (1≤ai≤1081≤𝑎𝑖≤108).On the third line, there are n𝑛 integers — the numbers b1,b2,…,bn𝑏1,𝑏2,…,𝑏𝑛 (1≤bi≤1081≤𝑏𝑖≤108).On the fourth line, there are n𝑛 integers — the numbers c1,c2,…,cn𝑐1,𝑐2,…,𝑐𝑛 (1≤ci≤1091≤𝑐𝑖≤109).On the fifth line, there are q𝑞 integers — the numbers d1,d2,…,dq𝑑1,𝑑2,…,𝑑𝑞 (0≤di≤10150≤𝑑𝑖≤1015).OutputPrint q𝑞 integers — the maximum value you can get for each of the q𝑞 possible values d𝑑.ExamplesinputCopy3 41 2 34 5 61 1 10 1 2 3outputCopy2 3 3 3 inputCopy5 53 4 6 8 48 3 4 9 310 20 30 40 505 55 13 1000 113outputCopy2 7 3 7 7 inputCopy1 13450outputCopy7