Problem: Write a Java program to find the lcm of an array (i.e. lcm of more than 2 numbers).

Example:

Input:  [8, 2, 4, 16]
Output: 16

Input:  [21, 7, 6]
Output: 42

To find the lcm of an array we will use GCD because, for any two numbers, the following relationship holds true:

LCM(n1, n2)= (n1*n2)/GCD(n1, n2)

But this is not valid for more than 2 numbers:

LCM(n1, n2, n3)= (n1*n2*n3)/GCD(n1, n2, n3)

Thus to find lcm of the whole array (more than 2 numbers), we will deal with two numbers at a time and use its result for the next number.

We will loop through the array and use Lcm of previous two numbers * array[n] / GCD as follows:

import java.util.*;

class Main {
  public static void main(String[] args) {
    int array[] = {8, 4, 2, 16};
    
    //initialize LCM and GCD with the first element
    int lcm = array[0];
    int gcd = array[0];
 
    //loop through the array to find GCD
    //use GCD to find the LCM
    for(int i=1; i<array.length; i++){
      gcd = findGCD(array[i], lcm);
      lcm = (lcm*array[i])/gcd;
    }
    
    //output the LCM
    System.out.println("LCM: "+lcm);
  }
 
  //recursive function to find GCD of two numbers
  public static int findGCD(int a, int b){
    //base condition
    if(b == 0)
      return a;
    
    return findGCD(b, a%b);
  }
}

Output:

LCM: 16

Explanation:

Suppose LCM of the first two numbers of the array (i.e, array[0] and array[1]) is lcm1, for next iteration we find the LCM of lcm1 and array[2] as lcm2, then for the 3rd iteration, LCM of lcm2 and array[3] as lcm3 and so on.

The last LCM value (i.e. LCM of lcm(n-1) and array[n]) gives the LCM of the whole array.

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