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Online Queries for GCD of array after divide operations

You are given an array of N integer values and M update operations. An update consists of choosing an element of the array and dividing it by a given value. It is guaranteed that the element is divisible by the chosen value. After each update, you should compute the greatest common divisor of all the elements of the array.

Examples:

```Input : 3 3
36 24 72
1 3
3 12
2 4
Output :12
6
6
After each operation the array values will be:
1. 12, 24, 72
2. 12, 24, 6
3. 12, 6, 6

Input :5 6
100 150 200 600 300
4 6
2 3
4 4
1 4
2 5
5 25
Output : 50
50
25
25
5
1```

Approach: First, you should compute the Greatest Common Divisor(gcd) of all the initial numbers. Because the queries consist of dividing a number by one of its divisors it means that after each query the new gcd is a divisor of the old gcd. So for each query, you should simply compute the gcd between the updated value and the previous gcd.

Implementation:

C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;` `void` `print_gcd_online(``int` `n, ``int` `m,``                      ``int` `query[][2], ``int` `arr[])``{``    ``// stores the gcd of the initial array elements``    ``int` `max_gcd = 0;``    ``int` `i = 0;` `    ``// calculates the gcd``    ``for` `(i = 0; i < n; i++)``        ``max_gcd = __gcd(max_gcd, arr[i]);` `    ``// performing online queries``    ``for` `(i = 0; i < m; i++)``    ``{``        ``// index is 1 based``        ``query[i][0]--;` `        ``// divide the array element``        ``arr[query[i][0]] /= query[i][1];` `        ``// calculates the current gcd``        ``max_gcd = __gcd(arr[query[i][0]], max_gcd);` `        ``// print the gcd after each step``        ``cout << max_gcd << endl;``    ``}``}` `// Driver code``int` `main()``{``    ``int` `n = 3;``    ``int` `m = 3;``    ``int` `query[m][2];``    ``int` `arr[] = {36, 24, 72};``    ``query[0][0] = 1;``    ``query[0][1] = 3;``    ``query[1][0] = 3;``    ``query[1][1] = 12;``    ``query[2][0] = 2;``    ``query[2][1] = 4;` `    ``print_gcd_online(n, m, query, arr);``    ``return` `0;``}` `// This code is contributed by``// sanjeev2552`

Java

 `// Java implementation of the approach` `class` `GFG {` `    ``// returns the gcd after all updates``    ``// in the array``    ``static` `int` `gcd(``int` `a, ``int` `b)``    ``{``        ``if` `(a == ``0``)``            ``return` `b;` `        ``return` `gcd(b % a, a);``    ``}` `    ``static` `void` `print_gcd_online(``int` `n, ``int` `m,``                    ``int``[][] query, ``int``[] arr)``    ``{` `        ``// stores the gcd of the initial array elements``        ``int` `max_gcd = ``0``;` `        ``int` `i = ``0``;``        ``for` `(i = ``0``; i < n; i++) ``// calculates the gcd``            ``max_gcd = gcd(max_gcd, arr[i]);` `        ``// performing online queries``        ``for` `(i = ``0``; i < m; i++) {` `            ``query[i][``0``]--; ``// index is 1 based` `            ``// divide the array element``            ``arr[query[i][``0``]] /= query[i][``1``];`` ` `            ``// calculates the current gcd``            ``max_gcd = gcd(arr[query[i][``0``]], max_gcd);` `            ``// print the gcd after each step``            ``System.out.println(max_gcd);``        ``}``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `n = ``3``;``        ``int` `m = ``3``;``        ``int``[][] query = ``new` `int``[m][``2``];``        ``int``[] arr = ``new` `int``[] { ``36``, ``24``, ``72` `};``        ``query[``0``][``0``] = ``1``;``        ``query[``0``][``1``] = ``3``;``        ``query[``1``][``0``] = ``3``;``        ``query[``1``][``1``] = ``12``;``        ``query[``2``][``0``] = ``2``;``        ``query[``2``][``1``] = ``4``;` `        ``print_gcd_online(n, m, query, arr);``    ``}``}`

Python3

 `# Python3 implementation of the``# above approach` `# Returns the gcd after all``# updates in the array``def` `gcd(a, b):``    ` `    ``if` `a ``=``=` `0``:``        ``return` `b` `    ``return` `gcd(b ``%` `a, a)` `def` `print_gcd_online(n, m, query, arr):` `    ``# Stores the gcd of the initial``    ``# array elements``    ``max_gcd ``=` `0` `    ``for` `i ``in` `range``(``0``, n): ``# calculates the gcd``        ``max_gcd ``=` `gcd(max_gcd, arr[i])` `    ``# performing online queries``    ``for` `i ``in` `range``(``0``, m):` `        ``query[i][``0``] ``-``=` `1` `# index is 1 based` `        ``# divide the array element``        ``arr[query[i][``0``]] ``/``/``=` `query[i][``1``]``    ` `        ``# calculates the current gcd``        ``max_gcd ``=` `gcd(arr[query[i][``0``]], max_gcd)` `        ``# Print the gcd after each step``        ``print``(max_gcd)` `# Driver code``if` `__name__ ``=``=` `"__main__"``:``    ` `    ``n, m ``=` `3``, ``3``    ``query ``=` `[[``1``,``3``], [``3``,``12``], [``2``,``4``]]``    ``arr ``=` `[``36``, ``24``, ``72``]``        ` `    ``print_gcd_online(n, m, query, arr)``    ` `# This code is contributed by Rituraj Jain`

C#

 `// C# implementation of the approach``using` `System;` `class` `GFG``{` `// returns the gcd after all``// updates in the array``static` `int` `gcd(``int` `a, ``int` `b)``{``    ``if` `(a == 0)``        ``return` `b;` `    ``return` `gcd(b % a, a);``}` `static` `void` `print_gcd_online(``int` `n, ``int` `m,``                             ``int``[,] query,``                             ``int``[] arr)``{` `    ``// stores the gcd of the``    ``// initial array elements``    ``int` `max_gcd = 0;` `    ``int` `i = 0;``    ``for` `(i = 0; i < n; i++) ``// calculates the gcd``        ``max_gcd = gcd(max_gcd, arr[i]);` `    ``// performing online queries``    ``for` `(i = 0; i < m; i++)``    ``{` `        ``query[i,0]--; ``// index is 1 based` `        ``// divide the array element``        ``arr[query[i, 0]] /= query[i, 1];` `        ``// calculates the current gcd``        ``max_gcd = gcd(arr[query[i, 0]], max_gcd);` `        ``// print the gcd after each step``        ``Console.WriteLine(max_gcd);``    ``}``}` `// Driver code``public` `static` `void` `Main()``{``    ``int` `n = 3;``    ``int` `m = 3;``    ``int``[,] query = ``new` `int``[m, 2];``    ``int``[] arr = ``new` `int``[] { 36, 24, 72 };``    ``query[0, 0] = 1;``    ``query[0, 1] = 3;``    ``query[1, 0] = 3;``    ``query[1, 1] = 12;``    ``query[2, 0] = 2;``    ``query[2, 1] = 4;` `    ``print_gcd_online(n, m, query, arr);``}``}` `// This code is contributed``// by Subhadeep Gupta`

PHP

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Javascript

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Output

```12
6
6```

Complexity Analysis:

• Time Complexity : O(m + n)
• Auxiliary Space: O(1)