# Maximum possible GCD after replacing at most one element in the given array

Given an array arr[] of size N > 1. The task is to find the maximum possible GCD of the array by replacing at most one element.

Examples:

Input: arr[] = {6, 7, 8}
Output: 2
Replace 7 with 2 and gcd(6, 2, 8) = 2
which is maximum possible.

Input: arr[] = {12, 18, 30}
Output: 6

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• Idea is to find the GCD value of all the sub-sequences of length (N – 1) and removing the element which has to be replaced in the sub-sequence as it can be replaced with any other element already in the subsequence. 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. Also note that suffixGCD[1] and prefixGCD[N – 2] also need to be compared in case the first or the last element has to be replaced.

Below is the implementation of the above approach:

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return the maximum ` `// possible gcd after replacing ` `// a single element ` `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[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 ` `    ``// index 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 replaced ` `    ``int` `ans = max(Suffix[2], 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[] = { 6, 7, 8 }; ` `    ``int` `n = ``sizeof``(a) / ``sizeof``(a[0]); ` ` `  `    ``cout << MaxGCD(a, n); ` ` `  `    ``return` `0; ` `} `

 `// Java implementation of the approach ` `class` `GFG  ` `{ ` ` `  `// Function to return the maximum ` `// possible gcd after replacing ` `// a single element ` `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 ` `    ``// index 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 replaced ` `    ``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; ` `} ` ` `  `static` `int` `__gcd(``int` `a, ``int` `b)  ` `{  ` `    ``return` `b == ``0` `? a : __gcd(b, a % b);      ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args)  ` `{ ` `    ``int` `a[] = { ``6``, ``7``, ``8` `}; ` `    ``int` `n = a.length; ` ` `  `    ``System.out.println(MaxGCD(a, n)); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992 `

 `# Python3 implementation of the approach  ` `from` `math ``import` `gcd as __gcd ` ` `  `# Function to return the maximum  ` `# possible gcd after replacing  ` `# a single element  ` `def` `MaxGCD(a, n) : ` ` `  `    ``# Prefix and Suffix arrays  ` `    ``Prefix ``=` `[``0``] ``*` `(n ``+` `2``);  ` `    ``Suffix ``=` `[``0``] ``*` `(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] ``=` `__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  ` `    ``# index greater than or equal to i in Suffix[i]  ` `    ``for` `i ``in` `range``(n ``-` `1``, ``0``, ``-``1``) : ` `        ``Suffix[i] ``=` `__gcd(Suffix[i ``+` `1``], a[i ``-` `1``]);  ` ` `  `    ``# If first or last element of  ` `    ``# the array has to be replaced  ` `    ``ans ``=` `max``(Suffix[``2``], Prefix[n ``-` `1``]);  ` ` `  `    ``# If any other element is replaced  ` `    ``for` `i ``in` `range``(``2``, n) : ` `        ``ans ``=` `max``(ans, __gcd(Prefix[i ``-` `1``],  ` `                             ``Suffix[i ``+` `1``]));  ` ` `  `    ``# Return the maximized gcd  ` `    ``return` `ans;  ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"` `: ` ` `  `    ``a ``=` `[ ``6``, ``7``, ``8` `];  ` `    ``n ``=` `len``(a);  ` ` `  `    ``print``(MaxGCD(a, n));  ` ` `  `# This code is contributed by AnkitRai01 `

 `// C# implementation of the approach ` `using` `System; ` `     `  `class` `GFG  ` `{ ` ` `  `// Function to return the maximum ` `// possible gcd after replacing ` `// a single element ` `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 ` `    ``// index 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 replaced ` `    ``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; ` `} ` ` `  `static` `int` `__gcd(``int` `a, ``int` `b)  ` `{  ` `    ``return` `b == 0 ? a : __gcd(b, a % b);      ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args)  ` `{ ` `    ``int` `[]a = { 6, 7, 8 }; ` `    ``int` `n = a.Length; ` ` `  `    ``Console.WriteLine(MaxGCD(a, n)); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:
```2
```

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

pawanasipugmailcom

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :