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# Remove an element to maximize the GCD of the given array

• Difficulty Level : Hard
• Last Updated : 07 May, 2021

Given an array arr[] of length N ≥ 2. The task is to remove an element from the given array such that the GCD of the array after removing it is maximized.

Examples:

Input: arr[] = {12, 15, 18}
Output:
Remove 12: GCD(15, 18) = 3
Remove 15: GCD(12, 18) = 6
Remove 18: GCD(12, 15) = 3

Input: arr[] = {14, 17, 28, 70}
Output: 14

Approach:

• Idea is to find the GCD value of all the sub-sequences of length (N – 1) and removing the element which is not present in the sub-sequence with that GCD. The maximum GCD found would be the answer.
• To find the GCD of the sub-sequences optimally, maintain a prefixGCD[] and a suffixGCD[] array using single state dynamic programming.
• The maximum value of GCD(prefixGCD[i – 1], suffixGCD[i + 1]) is the required answer.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the above approach``#include ``using` `namespace` `std;` `// Function to return the maximized gcd``// after removing a single element``// from the given array``int` `MaxGCD(``int` `a[], ``int` `n)``{` `    ``// Prefix and Suffix arrays``    ``int` `Prefix[n + 2];``    ``int` `Suffix[n + 2];` `    ``// Single state dynamic programming relation``    ``// for storing gcd of first i elements``    ``// from the left in Prefix[i]``    ``Prefix = a;``    ``for` `(``int` `i = 2; i <= n; i += 1) {``        ``Prefix[i] = __gcd(Prefix[i - 1], a[i - 1]);``    ``}` `    ``// Initializing Suffix array``    ``Suffix[n] = a[n - 1];` `    ``// Single state dynamic programming relation``    ``// for storing gcd of all the elements having``    ``// greater than or equal to i in Suffix[i]``    ``for` `(``int` `i = n - 1; i >= 1; i -= 1) {``        ``Suffix[i] = __gcd(Suffix[i + 1], a[i - 1]);``    ``}` `    ``// If first or last element of``    ``// the array has to be removed``    ``int` `ans = max(Suffix, Prefix[n - 1]);` `    ``// If any other element is replaced``    ``for` `(``int` `i = 2; i < n; i += 1) {``        ``ans = max(ans, __gcd(Prefix[i - 1], Suffix[i + 1]));``    ``}` `    ``// Return the maximized gcd``    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``int` `a[] = { 14, 17, 28, 70 };``    ``int` `n = ``sizeof``(a) / ``sizeof``(a);` `    ``cout << MaxGCD(a, n);` `    ``return` `0;``}`

## Java

 `// Java implementation of the above approach``class` `Test``{``    ``// Recursive function to return gcd of a and b``    ``static` `int` `gcd(``int` `a, ``int` `b)``    ``{``        ``if` `(b == ``0``)``            ``return` `a;``        ``return` `gcd(b, a % b);``    ``}``    ` `    ``// Function to return the maximized gcd``    ``// after removing a single element``    ``// from the given array``    ``static` `int` `MaxGCD(``int` `a[], ``int` `n)``    ``{``    ` `        ``// Prefix and Suffix arrays``        ``int` `Prefix[] = ``new` `int``[n + ``2``];``        ``int` `Suffix[] = ``new` `int``[n + ``2``] ;``    ` `        ``// Single state dynamic programming relation``        ``// for storing gcd of first i elements``        ``// from the left in Prefix[i]``        ``Prefix[``1``] = a[``0``];``        ``for` `(``int` `i = ``2``; i <= n; i += ``1``)``        ``{``            ``Prefix[i] = gcd(Prefix[i - ``1``], a[i - ``1``]);``        ``}``    ` `        ``// Initializing Suffix array``        ``Suffix[n] = a[n - ``1``];``    ` `        ``// Single state dynamic programming relation``        ``// for storing gcd of all the elements having``        ``// greater than or equal to i in Suffix[i]``        ``for` `(``int` `i = n - ``1``; i >= ``1``; i -= ``1``)``        ``{``            ``Suffix[i] = gcd(Suffix[i + ``1``], a[i - ``1``]);``        ``}``    ` `        ``// If first or last element of``        ``// the array has to be removed``        ``int` `ans = Math.max(Suffix[``2``], Prefix[n - ``1``]);``    ` `        ``// If any other element is replaced``        ``for` `(``int` `i = ``2``; i < n; i += ``1``)``        ``{``            ``ans = Math.max(ans, gcd(Prefix[i - ``1``], Suffix[i + ``1``]));``        ``}``    ` `        ``// Return the maximized gcd``        ``return` `ans;``    ``}``        ` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{` `        ``int` `a[] = { ``14``, ``17``, ``28``, ``70` `};``        ``int` `n = a.length;``    ` `        ``System.out.println(MaxGCD(a, n));``    ``}``}` `// This code is contributed by AnkitRai01`

## Python3

 `# Python3 implementation of the above approach``import` `math as mt` `# Function to return the maximized gcd``# after removing a single element``# from the given array` `def` `MaxGCD(a, n):`  `    ``# Prefix and Suffix arrays``    ``Prefix``=``[``0` `for` `i ``in` `range``(n ``+` `2``)]``    ``Suffix``=``[``0` `for` `i ``in` `range``(n ``+` `2``)]` `    ``# Single state dynamic programming relation``    ``# for storing gcd of first i elements``    ``# from the left in Prefix[i]``    ``Prefix[``1``] ``=` `a[``0``]``    ``for` `i ``in` `range``(``2``,n``+``1``):``        ``Prefix[i] ``=` `mt.gcd(Prefix[i ``-` `1``], a[i ``-` `1``])` `    ``# Initializing Suffix array``    ``Suffix[n] ``=` `a[n ``-` `1``]` `    ``# Single state dynamic programming relation``    ``# for storing gcd of all the elements having``    ``# greater than or equal to i in Suffix[i]``    ``for` `i ``in` `range``(n``-``1``,``0``,``-``1``):``        ``Suffix[i] ``=``mt.gcd(Suffix[i ``+` `1``], a[i ``-` `1``])` `    ``# If first or last element of``    ``# the array has to be removed``    ``ans ``=` `max``(Suffix[``2``], Prefix[n ``-` `1``])` `    ``# If any other element is replaced``    ``for` `i ``in` `range``(``2``,n):``        ``ans ``=` `max``(ans, mt.gcd(Prefix[i ``-` `1``], Suffix[i ``+` `1``]))` `    ``# Return the maximized gcd``    ``return` `ans` `# Driver code` `a``=``[``14``, ``17``, ``28``, ``70``]``n ``=` `len``(a)` `print``(MaxGCD(a, n))` `# This code is contributed by mohit kumar 29`

## C#

 `// C# implementation of the above approach``using` `System;` `class` `GFG``{``    ` `    ``// Recursive function to return gcd of a and b``    ``static` `int` `gcd(``int` `a, ``int` `b)``    ``{``        ``if` `(b == 0)``            ``return` `a;``        ``return` `gcd(b, a % b);``    ``}``    ` `    ``// Function to return the maximized gcd``    ``// after removing a single element``    ``// from the given array``    ``static` `int` `MaxGCD(``int` `[]a, ``int` `n)``    ``{``    ` `        ``// Prefix and Suffix arrays``        ``int` `[]Prefix = ``new` `int``[n + 2];``        ``int` `[]Suffix = ``new` `int``[n + 2] ;``    ` `        ``// Single state dynamic programming relation``        ``// for storing gcd of first i elements``        ``// from the left in Prefix[i]``        ``Prefix = a;``        ``for` `(``int` `i = 2; i <= n; i += 1)``        ``{``            ``Prefix[i] = gcd(Prefix[i - 1], a[i - 1]);``        ``}``    ` `        ``// Initializing Suffix array``        ``Suffix[n] = a[n - 1];``    ` `        ``// Single state dynamic programming relation``        ``// for storing gcd of all the elements having``        ``// greater than or equal to i in Suffix[i]``        ``for` `(``int` `i = n - 1; i >= 1; i -= 1)``        ``{``            ``Suffix[i] = gcd(Suffix[i + 1], a[i - 1]);``        ``}``    ` `        ``// If first or last element of``        ``// the array has to be removed``        ``int` `ans = Math.Max(Suffix, Prefix[n - 1]);``    ` `        ``// If any other element is replaced``        ``for` `(``int` `i = 2; i < n; i += 1)``        ``{``            ``ans = Math.Max(ans, gcd(Prefix[i - 1], Suffix[i + 1]));``        ``}``    ` `        ``// Return the maximized gcd``        ``return` `ans;``    ``}``        ` `    ``// Driver code``    ``static` `public` `void` `Main ()``    ``{``        ` `        ``int` `[]a = { 14, 17, 28, 70 };``        ``int` `n = a.Length;``    ` `        ``Console.Write(MaxGCD(a, n));``    ``}``}` `// This code is contributed by ajit.`

## Javascript

 ``
Output:
`14`

Time Complexity: O(N * log(M)) where M is the maximum element from the array.

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