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# Maximize K to make given array Palindrome when each element is replaced by its remainder with K

• Last Updated : 12 Jul, 2021

Given an array A[] containing N positive integers, the task is to find the largest possible number K, such that after replacing all elements by the elements modulo K(A[i]=A[i]%K, for all 0<=i<N), the array becomes a palindrome. If K is infinitely large, print -1.

Examples:

Input: A={1, 2, 3, 4}, N=4
Output:
1
Explanation:
For K=1, A becomes {1%1, 2%1, 3%1, 4%1}={0, 0, 0, 0} which is a palindromic array.

Input: A={1, 2, 3, 2, 1}, N=5
Output:
-1

Observation: The following observations help in solving the problem:

1. If the array is already a palindrome, K can be infinitely large.
2. Two numbers, say A and B can be made equal by taking their modulus with their difference(|A-B|) as well as the factors of their difference.

Approach: The problem can be solved by making K equal to the GCD of the absolute differences of A[i] and A[N-i-1]. Follow the steps below to solve the problem:

1. Check whether A is already a palindrome. If it is, return -1.
2. Store the absolute difference of the first and last elements of the array in a variable, say K, which will store the largest number required to change A into a palindrome.
3. Traverse from 1 to N/2-1, and for each current index i, do the following:
1. Update K with the GCD of K and the absolute difference of A[i] and A[N-i-1].
4. Return K.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// utility function to calculate the GCD of two numbers``int` `gcd(``int` `a, ``int` `b)``{``    ``if` `(b == 0)``        ``return` `a;``    ``else``        ``return` `gcd(b, a % b);``}``// Function to calculate the largest K, replacing all``// elements of an array A by their modulus with K, makes A a``// palindromic array``int` `largestK(``int` `A[], ``int` `N)``{``    ``// check if A is palindrome``    ``int` `l = 0, r = N - 1, flag = 0;``    ``while` `(l < r) {``        ``// A is not palindromic``        ``if` `(A[l] != A[r]) {``            ``flag = 1;``            ``break``;``        ``}``        ``l++;``        ``r--;``    ``}``    ``// K can be infitely large in this case``    ``if` `(flag == 0)``        ``return` `-1;` `    ``// variable to store the largest K that makes A``    ``// palindromic``    ``int` `K = ``abs``(A[0] - A[N - 1]);``    ``for` `(``int` `i = 1; i < N / 2; i++)``        ``K = gcd(K, ``abs``(A[i] - A[N - i - 1]));``    ``// return the required answer``    ``return` `K;``}``// Driver code``int` `main()``{``    ``// Input``    ``int` `A[] = { 1, 2, 3, 2, 1 };``    ``int` `N = ``sizeof``(A) / ``sizeof``(A[0]);` `    ``// Function call``    ``cout << largestK(A, N) << endl;``    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.io.*;` `class` `GFG{` `// Utility function to calculate the GCD``// of two numbers``static` `int` `gcd(``int` `a, ``int` `b)``{``    ``if` `(b == ``0``)``        ``return` `a;``    ``else``        ``return` `gcd(b, a % b);``}` `// Function to calculate the largest K,``// replacing all elements of an array A``// by their modulus with K, makes A a``// palindromic array``static` `int` `largestK(``int` `A[], ``int` `N)``{``    ` `    ``// Check if A is palindrome``    ``int` `l = ``0``, r = N - ``1``, flag = ``0``;``    ``while` `(l < r)``    ``{``        ` `        ``// A is not palindromic``        ``if` `(A[l] != A[r])``        ``{``            ``flag = ``1``;``            ``break``;``        ``}``        ``l++;``        ``r--;``    ``}``    ` `    ``// K can be infitely large in this case``    ``if` `(flag == ``0``)``        ``return` `-``1``;` `    ``// Variable to store the largest K``    ``// that makes A palindromic``    ``int` `K = Math.abs(A[``0``] - A[N - ``1``]);``    ``for``(``int` `i = ``1``; i < N / ``2``; i++)``        ``K = gcd(K, Math.abs(A[i] - A[N - i - ``1``]));``        ` `    ``// Return the required answer``    ``return` `K;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ` `    ``// Input``    ``int` `A[] = { ``1``, ``2``, ``3``, ``2``, ``1` `};``    ``int` `N = A.length;``    ` `    ``// Function call``    ``System.out.println(largestK(A, N));``}``}` `// This code is contributed by sanjoy_62`

## Python3

 `# Python3 program for the above approach` `# utility function to calculate the GCD of two numbers``def` `gcd(a, b):``    ``if` `(b ``=``=` `0``):``        ``return` `a``    ``else``:``        ``return` `gcd(b, a ``%` `b)``      ` `# Function to calculate the largest K, replacing all``# elements of an array A by their modulus with K, makes A a``# palindromic array``def` `largestK(A, N):``  ` `    ``# check if A is palindrome``    ``l,r,flag ``=` `0``, N ``-` `1``, ``0``    ``while` `(l < r):``        ``# A is not palindromic``        ``if` `(A[l] !``=` `A[r]):``            ``flag ``=` `1``            ``break``        ``l ``+``=` `1``        ``r ``-``=` `1``    ``# K can be infitely large in this case``    ``if` `(flag ``=``=` `0``):``        ``return` `-``1` `    ``# variable to store the largest K that makes A``    ``# palindromic``    ``K ``=` `abs``(A[``0``] ``-` `A[N ``-` `1``])``    ``for` `i ``in` `range``(``1``,N``/``/``2``):``        ``K ``=` `gcd(K, ``abs``(A[i] ``-` `A[N ``-` `i ``-` `1``]))``    ` `    ``# return the required answer``    ``return` `K``  ` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``# Input``    ``A``=` `[``1``, ``2``, ``3``, ``2``, ``1` `]``    ``N ``=` `len``(A)` `    ``# Function call``    ``print` `(largestK(A, N))` `# This code is contributed by mohit kumar 29.`

## C#

 `// c# program for the above approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG{` `// utility function to calculate the GCD of two numbers``static` `int` `gcd(``int` `a, ``int` `b)``{``    ``if` `(b == 0)``        ``return` `a;``    ``else``        ``return` `gcd(b, a % b);``}``  ` `// Function to calculate the largest K, replacing all``// elements of an array A by their modulus with K, makes A a``// palindromic array``static` `int` `largestK(``int` `[]A, ``int` `N)``{``  ` `    ``// check if A is palindrome``    ``int` `l = 0, r = N - 1, flag = 0;``    ``while` `(l < r) {``        ``// A is not palindromic``        ``if` `(A[l] != A[r]) {``            ``flag = 1;``            ``break``;``        ``}``        ``l++;``        ``r--;``    ``}``  ` `    ``// K can be infitely large in this case``    ``if` `(flag == 0)``        ``return` `-1;` `    ``// variable to store the largest K that makes A``    ``// palindromic``    ``int` `K = Math.Abs(A[0] - A[N - 1]);``    ``for` `(``int` `i = 1; i < N / 2; i++)``        ``K = gcd(K, Math.Abs(A[i] - A[N - i - 1]));``  ` `    ``// return the required answer``    ``return` `K;``}` `// Driver code``public` `static` `void` `Main()``{``  ` `    ``// Input``    ``int` `[]A = { 1, 2, 3, 2, 1 };``    ``int` `N = A.Length;` `    ``// Function call``    ``Console.Write(largestK(A, N));``}``}` `// This code is contributed by ipg2016107.`

## Javascript

 ``
Output
`-1`

Time Complexity: O(NLogM), where M is the largest element in the array
Auxiliary Space: O(1)

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