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Construct longest possible sequence of unique elements with given LCM
  • Last Updated : 05 Jan, 2021

Given a positive integer N, the task is to construct the longest sorted sequence of unique elements whose LCM of the is equal to N.

Examples:

Input: N = 12 
Output: 1 2 3 4 6 12 
Explanation: 
LCM of {1, 2, 3, 4, 6, 12 } is N( = 12). 
Therefore, the longest possible sequence is {1, 2, 3, 4, 6, 12 }.

Input: N = 9 
Output: 1 3 9 
Explanation: 
LCM of { 1, 2, 9 } is N( = 9). 
Therefore, the longest possible sequence is {1, 3, 9 }.

Approach: The problem can be solved based on the following observation:



If an array element is not a factor of N then the LCM of the array elements never be N. Therefore, the array elements must be the factor of N.

Follow the below steps to solve this problem:

Below is the implementation of the above approach:

C++




// C++ program to implement
// the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to construct an array of
// unique elements whose LCM is N
void constructArrayWithGivenLCM(int N)
{
    // Stores array elements
    // whose LCM is N
    vector<int> newArr;
 
    // Iterate over the range
    // [1, sqrt(N)]
    for (int i = 1; i * i <= N;
         i++) {
 
        // If N is divisible
        // by i
        if (N % i == 0) {
 
            // Insert i into newArr[]
            newArr.push_back(i);
 
            // If N is not perfect square
            if (N / i != i) {
                newArr.push_back(N / i);
            }
        }
    }
 
    // Sort the array newArr[]
    sort(newArr.begin(), newArr.end());
 
    // Print array elements
    for (auto i : newArr) {
 
        cout << i << " ";
    }
}
 
// Driver Code
int main()
{
    // Given N
    int N = 12;
 
    // Function Call
    constructArrayWithGivenLCM(N);
 
    return 0;
}

Java




// Java program to implement
// the above approach
import java.util.Arrays;
 
class GFG{
 
// Function to construct an array of
// unique elements whose LCM is N
static void constructArrayWithGivenLCM(int N)
{
     
    // Stores array elements
    // whose LCM is N
    int newArr[] = new int[N];
 
    int j = 0;
     
    // Iterate over the range
    // [1, sqrt(N)]
    for(int i = 1; i * i <= N; i++)
    {
         
        // If N is divisible
        // by i
        if (N % i == 0)
        {
             
            // Insert i into newArr[]
            newArr[j] = i;
            j++;
 
            // If N is not perfect square
            if (N / i != i)
            {
                newArr[j] = N / i;
                j++;
            }
        }
    }
 
    // Sort the array newArr[]
    Arrays.sort(newArr);
 
    // Print array elements
    for(int i = j; i < N; i++) 
    {
        System.out.print(newArr[i] + " ");
    }
}
 
// Driver Code
public static void main (String[] args)
{
     
    // Given N
    int N = 12;
     
    // Function Call
    constructArrayWithGivenLCM(N);
}
}
 
// This code is contributed by AnkThon

Python3




# Python3 program to implement
# the above approach
from math import sqrt,ceil,floor
 
# Function to construct an array of
# unique elements whose LCM is N
def constructArrayWithGivenLCM(N):
   
    # Stores array elements
    # whose LCM is N
    newArr = []
 
    # Iterate over the range
    # [1, sqrt(N)]
    for i in range(1, ceil(sqrt(N + 1))):
 
        # If N is divisible
        # by i
        if (N % i == 0):
 
            # Insert i into newArr[]
            newArr.append(i)
 
            # If N is not perfect square
            if (N // i != i):
                newArr.append(N // i)
 
    # Sort the array newArr[]
    newArr = sorted(newArr)
 
    # Print array elements
    for i in newArr:
        print(i, end = " ")
 
# Driver Code
if __name__ == '__main__':
 
  # Given N
    N = 12
 
    # Function Call
    constructArrayWithGivenLCM(N)
 
# This code is contributed by mohit kumar 29

C#




// C# program to implement
// the above approach
using System;
class GFG{
 
  // Function to construct an array of
  // unique elements whose LCM is N
  static void constructArrayWithGivenLCM(int N)
  {
 
    // Stores array elements
    // whose LCM is N
    int []newArr = new int[N];
 
    int j = 0;
 
    // Iterate over the range
    // [1, sqrt(N)]
    for(int i = 1; i * i <= N; i++)
    {
 
      // If N is divisible
      // by i
      if (N % i == 0)
      {
 
        // Insert i into newArr[]
        newArr[j] = i;
        j++;
 
        // If N is not perfect square
        if (N / i != i)
        {
          newArr[j] = N / i;
          j++;
        }
      }
    }
 
    // Sort the array newArr[]
    Array.Sort(newArr);
 
    // Print array elements
    for(int i = j; i < N; i++) 
    {
      Console.Write(newArr[i] + " ");
    }
  }
 
  // Driver Code
  public static void Main(String[] args)
  {
 
    // Given N
    int N = 12;
 
    // Function Call
    constructArrayWithGivenLCM(N);
  }
}
 
// This code is contributed by Princi Singh
Output: 
1 2 3 4 6 12

 

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

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