Maximize the sum of modulus with every Array element

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:

  1. Calcaulate the LCM of all array elements.
  2. 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 ith index.
  3. 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 ith index, which is the maximum possible.
  4. Hence, the maximum possible value of F(M) can be Sum of array elements – N.

Below is the implementation of the above approach:

C++

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// 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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output:

10

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

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