Maximize the sum of modulus with every Array element

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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#

// 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)

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