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. Calculate 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 i^th 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 i^th 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++

// 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:

10

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