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 ith 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 ith 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 elementint 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 Programint 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// elementimport java.io.*; class GFG{ // Function to return the maximum// sum of modulus with every array// elementstatic 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 elementdef 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 Codea = [ 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// elementusing System;class GFG{ // Function to return the maximum// sum of modulus with every array// elementstatic 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)
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