Minimum LCM of all pairs in a given array

Given an array arr[] of size N, the task is to find the minimum LCM (Least Common Multiple) of all unique pairs in the given array, where 1 <= N <= 10⁵, 1 <= arr[i] <= 10⁵.

Examples:

Input: arr[] = {2, 4, 3}
Output: 4
Explanation
LCM (2, 4) = 4
LCM (2, 3) = 6
LCM (4, 3) = 12
Minimum possible LCM is 4.

Input: arr [] ={1, 5, 2, 2, 6}
Output: 2

Naive Approach

Generate all possible pairs and compute LCM for every unique pair.
Find the minimum LCM from all unique pairs.

Below is the implementation of the above approach:

C++

// C++ program to find
// minimum possible lcm
// from any pair

#include <bits/stdc++.h>
using namespace std;

// function to compute
// GCD of two numbers
int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}

// function that return
// minimum possible lcm
// from any pair
int minLCM(int arr[], int n)
{
    int ans = INT_MAX;
    for (int i = 0; i < n; i++) {

        // fix the ith element and
        // iterate over all the array
        // to find minimum LCM
        for (int j = i + 1; j < n; j++) {

            int g = gcd(arr[i], arr[j]);
            int lcm = arr[i] / g * arr[j];
            ans = min(ans, lcm);
        }
    }

    return ans;
}

// Driver code
int main()
{
    int arr[] = { 2, 4, 3, 6, 5 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << minLCM(arr, n) << endl;
    return 0;
}

Java

// Java program to find minimum
// possible lcm from any pair
import java.io.*; 
import java.util.*; 

class GFG { 
    
// Function to compute
// GCD of two numbers
static int gcd(int a, int b)
{ 
    if (b == 0)
        return a;
    return gcd(b, a % b);
} 

// Function that return minimum
// possible lcm from any pair
static int minLCM(int arr[], int n)
{
    int ans = Integer.MAX_VALUE;
    
    for(int i = 0; i < n; i++) 
    {
        
       // Fix the ith element and
       // iterate over all the array
       // to find minimum LCM
       for(int j = i + 1; j < n; j++) 
       {
          int g = gcd(arr[i], arr[j]);
          int lcm = arr[i] / g * arr[j];
          ans = Math.min(ans, lcm);
       }
    }
    return ans;
}
    
// Driver code 
public static void main(String[] args) 
{ 
    int arr[] = { 2, 4, 3, 6, 5 };
    int n = arr.length;
    
    System.out.println(minLCM(arr,n)); 
} 
} 

// This code is contributed by coder001

Python3

# Python3 program to find minimum 
# possible lcm from any pair 
import sys

# Function to compute 
# GCD of two numbers 
def gcd(a, b):
    
    if (b == 0):
        return a;
    return gcd(b, a % b);

# Function that return minimum 
# possible lcm from any pair
def minLCM(arr, n):
    
    ans = 1000000000;
    for i in range(n):

        # Fix the ith element and 
        # iterate over all the 
        # array to find minimum LCM
        for j in range(i + 1, n):
            
            g = gcd(arr[i], arr[j]);
            lcm = arr[i] / g * arr[j];
            ans = min(ans, lcm);
            
    return ans;

# Driver code
arr = [ 2, 4, 3, 6, 5 ];

print(minLCM(arr, 5))

# This code is contributed by grand_master

// C# program to find minimum
// possible lcm from any pair
using System;
class GFG{ 
    
// Function to compute
// GCD of two numbers
static int gcd(int a, int b)
{ 
    if (b == 0)
        return a;
    return gcd(b, a % b);
} 

// Function that return minimum
// possible lcm from any pair
static int minLCM(int []arr, int n)
{
    int ans = Int32.MaxValue;
    
    for(int i = 0; i < n; i++) 
    {
        
    // Fix the ith element and
    // iterate over all the array
    // to find minimum LCM
    for(int j = i + 1; j < n; j++) 
    {
        int g = gcd(arr[i], arr[j]);
        int lcm = arr[i] / g * arr[j];
        ans = Math.Min(ans, lcm);
    }
    }
    return ans;
}
    
// Driver code 
public static void Main() 
{ 
    int []arr = { 2, 4, 3, 6, 5 };
    int n = arr.Length;
    
    Console.Write(minLCM(arr,n)); 
} 
} 

// This code is contributed by Akanksha_Rai

JavaScript

<script>

    // Javascript program to find
    // minimum possible lcm
    // from any pair
    
    // function to compute
    // GCD of two numbers
    function gcd(a, b)
    {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }

    // function that return
    // minimum possible lcm
    // from any pair
    function minLCM(arr, n)
    {
        let ans = Number.MAX_VALUE;
        for (let i = 0; i < n; i++) {

            // fix the ith element and
            // iterate over all the array
            // to find minimum LCM
            for (let j = i + 1; j < n; j++) {

                let g = gcd(arr[i], arr[j]);
                let lcm = arr[i] / g * arr[j];
                ans = Math.min(ans, lcm);
            }
        }

        return ans;
    }
    
    let arr = [ 2, 4, 3, 6, 5 ];
    let n = arr.length;
    document.write(minLCM(arr, n));


</script>

Output:

Time Complexity: O(N²)
Auxiliary Space: O(log(A_i)), where A_i is the second maximum element in the array.

Efficient Approach: This approach depends upon the formula:

Product of two number = LCM of two number * GCD of two number

In the formula of LCM, the denominator is the GCD of two numbers, and the GCD of two numbers will never be greater than the number itself.
So for a fixed GCD, find the smallest two multiples of that fixed GCD that is present in the given array.
Store only the smallest two multiples of each GCD because choosing a bigger multiple of GCD that is present in the array, no matter what, it will never give the minimum answer.
Finally, use a sieve to find the minimum two number that is the multiple of the chosen GCD.