Maximize the sum of modulus with every Array element

Last Updated : 14 Jan, 2022

Given an array A[] consisting of N positive integers, the task is to find the maximum possible value of:

F(M) = M % A[0] + M % A[1] + .... + M % A[N -1] where M can be any integer value

Examples:

Input: arr[] = {3, 4, 6}
Output: 10
Explanation:
The maximum sum occurs for M = 11.
(11 % 3) + (11 % 4) + (11 % 6) = 2 + 3 + 5 = 10
Input: arr[] = {2, 5, 3}
Output:7
Explanation:
The maximum sum occurs for M = 29.
(29 % 2) + (29 % 5) + (29 % 3) = 1 + 4 + 2 = 7.

Approach:
Follow the steps below to solve the problem:

Calculate the LCM of all array elements.
If M is equal to the LCM of the array, then F(M) = 0 i.e. the minimum possible value of the F(M). This is because, M % a[i] will always be 0 for every i^th index.
For M = LCM of array elements - 1, F(M) is maximized. This is because, M % a[i] is equal to a[i] - 1 for every i^th index, which is the maximum possible.
Hence, the maximum possible value of F(M) can be Sum of array elements - N.

Below is the implementation of the above approach:

C++

// C++ program to find the // maximum sum of modulus // with every array element #include <bits/stdc++.h> using namespace std;  // Function to return the // maximum sum of modulus // with every array element int maxModulosum(int a[], int n) {     int sum = 0;      // Sum of array elements     for (int i = 0; i < n; i++) {         sum += a[i];     }      // Return the answer     return sum - n; }  // Driver Program int main() {     int a[] = { 3, 4, 6 };     int n = sizeof(a) / sizeof(a[0]);     cout << maxModulosum(a, n);      return 0; }

Java

// Java program to find the maximum // sum of modulus with every array // element import java.io.*;   class GFG{   // Function to return the maximum // sum of modulus with every array // element static int maxModulosum(int a[], int n) {     int sum = 0;          // Sum of array elements     for(int i = 0; i < n; i++)     {        sum += a[i];     }          // Return the answer     return sum - n; }      // Driver Code  public static void main (String[] args)  {      int a[] = new int[]{ 3, 4, 6 };     int n = a.length;          System.out.println(maxModulosum(a, n));  }  }   // This code is contributed by Shubham Prakash

Python3

# Python3 program to find the # maximum sum of modulus # with every array element  # Function to return the # maximum sum of modulus # with every array element def maxModulosum(a, n):      sum1 = 0;      # Sum of array elements     for i in range(0, n):         sum1 += a[i];          # Return the answer     return sum1 - n;  # Driver Code a = [ 3, 4, 6 ]; n = len(a); print(maxModulosum(a, n));  # This code is contributed by Code_Mech

// C# program to find the maximum // sum of modulus with every array // element using System; class GFG{   // Function to return the maximum // sum of modulus with every array // element static int maxModulosum(int []a, int n) {     int sum = 0;          // Sum of array elements     for(int i = 0; i < n; i++)     {         sum += a[i];     }          // Return the answer     return sum - n; }      // Driver Code  public static void Main(String[] args)  {      int []a = new int[]{ 3, 4, 6 };     int n = a.Length;          Console.Write(maxModulosum(a, n));  }  }   // This code is contributed  // by shivanisinghss2110

JavaScript

<script>      // Javascript program to find the      // maximum sum of modulus      // with every array element           // Function to return the      // maximum sum of modulus      // with every array element      function maxModulosum(a, n)      {          let sum = 0;           // Sum of array elements          for (let i = 0; i < n; i++) {              sum += a[i];          }           // Return the answer          return sum - n;      }            let a = [ 3, 4, 6 ];      let n = a.length;     document.write(maxModulosum(a, n));  </script>

Output:

Time Complexity: O(N)
Auxiliary Space: O(1)

Maximize the sum of Array by formed by adding pair of elements

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Maximize the sum of modulus with every Array element

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