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Group all co-prime numbers from 1 to N

  • Difficulty Level : Expert
  • Last Updated : 08 Nov, 2021

Given an integer N, the task is to group numbers such that each group is mutually co-prime together with the total grouping is minimum.

Examples:

Input: N = 8 
Output: 
1 2 3 
4 5 
6 7 
8

Input: N = 5 
Output: 
1 2 3 
4 5

Approach: The key observation in this problem is two consecutive numbers are always co-prime. That is GCD(a, a+1) = 1. Another important observation is even numbers can’t be listed in one group. Because they will lead to the greatest common divisor of 2. Therefore, every consecutive even and odd numbers can be grouped into one group and 1 can be in any group because the greatest common divisor of numbers with 1 is always 1.

Below is the implementation of the above approach :

C++




// C++ implementation to group
// mutually coprime numbers into
// one group with minimum group possible
#include<bits/stdc++.h>
using namespace std;
 
// Function to group the mutually
// co-prime numbers into one group
void mutually_coprime(int n)
{
    if (n <= 3)
    {
         
        // Loop for the numbers less
        // than the 4
        for(int j = 1; j <= n; j++)
        {
            cout << j << " ";
        }
        cout << "\n";
    }
    else
    {
         
        // Integers 1, 2 and 3 can be
        // grouped into one group
        cout << "1 2 3\n";
         
        for(int j = 4; j < n; j += 2)
        {
             
            // Consecutive even and
            // odd numbers
            cout << j << " " << j + 1 << "\n";
        }
        if(n % 2 == 0)
            cout << n << "\n";
    }
}
 
// Driver Code        
int main()
{
    int n = 9;
     
    // Function call
    mutually_coprime(n);
}
 
// This code is contributed by yatinagg

Java




// Java implementation to group
// mutually coprime numbers into
// one group with minimum group possible
class GFG{
     
// Function to group the mutually
// co-prime numbers into one group
static void mutually_coprime(int n)
{
    if (n <= 3)
    {
         
        // Loop for the numbers less
        // than the 4
        for(int j = 1; j < n + 1; j++)
           System.out.print(j + " ");
        System.out.println();
    }
    else
    {
         
        // Integers 1, 2 and 3 can be
        // grouped into one group
        System.out.println("1 2 3");
        for(int j = 4; j < n; j += 2)
        {
 
           // Consecutive even and
           // odd numbers
           System.out.println(j + " " + (j + 1));
           if (n % 2 == 0)
           System.out.println(n);
        }
    }
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 9;
 
    // Function Call
    mutually_coprime(n);
}
}
 
// This code is contributed by sapnasingh4991

Python3




# Python3 implementation to group
# mutually coprime numbers into
# one group with minimum group possible
 
# Function to group the mutually
# co-prime numbers into one group
def mutually_coprime (n):   
    if ( n <= 3):
        # Loop for the numbers less
        # than the 4
        for j in range (1, n + 1):
            print (j, end =" ")
        print ()
    else:
        # Integers 1, 2 and 3 can be
        # grouped into one group
        print (1, 2, 3)
        for j in range ( 4, n, 2 ):
             
            # Consecutive even and
            # odd numbers
            print (j, ( j + 1 ))
        if(n % 2 == 0):        
            print (n)
 
# Driver Code           
if __name__ == "__main__":
    n = 9
     
    # Function Call
    mutually_coprime (n)

C#




// C# implementation to group
// mutually coprime numbers into
// one group with minimum group possible
using System;
 
class GFG{
     
// Function to group the mutually
// co-prime numbers into one group
static void mutually_coprime(int n)
{
    if (n <= 3)
    {
         
        // Loop for the numbers less
        // than the 4
        for(int j = 1; j < n + 1; j++)
           Console.Write(j + " ");
            
        Console.WriteLine();
    }
    else
    {
         
        // ints 1, 2 and 3 can be
        // grouped into one group
        Console.WriteLine("1 2 3");
        for(int j = 4; j < n; j += 2)
        {
           // Consecutive even and
           // odd numbers
           Console.WriteLine(j + " " + (j + 1));
             
           if (n % 2 == 0)
               Console.WriteLine(n);
        }
    }
}
 
// Driver Code
public static void Main(String[] args)
{
    int n = 9;
 
    // Function Call
    mutually_coprime(n);
}
}
// This code is contributed by sapnasingh4991

Javascript




<script>
 
// Javascript implementation to group
// mutually coprime numbers into
// one group with minimum group possible
 
// Function to group the mutually
// co-prime numbers into one group
function mutually_coprime(n)
{
    if (n <= 3)
    {
           
        // Loop for the numbers less
        // than the 4
        for(let j = 1; j < n + 1; j++)
           document.write(j + " " + "<br/>");
        document.write("<br/>");
    }
    else
    {
           
        // Integers 1, 2 and 3 can be
        // grouped into one group
        document.write("1 2 3" + "<br/>");
        for(let j = 4; j < n; j += 2)
        {
   
           // Consecutive even and
           // odd numbers
           document.write(j + " " + (j + 1) + "<br/>");
           if (n % 2 == 0)
           document.write(n + "<br/>");
        }
    }
}
   
 
// Driver Code
     
    let n = 9;
   
    // Function Call
    mutually_coprime(n);
       
</script>
Output: 
1 2 3
4 5
6 7
8 9         

 

Time Complexity: O(n)

Auxiliary Space: O(1)


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