# Split first N natural numbers into two subsequences with non-coprime sums

• Difficulty Level : Medium
• Last Updated : 23 Apr, 2021

Given an integer N (N &e; 3), the task is to split all numbers from 1 to N into two subsequences such that the sum of two subsequences is non-coprime to each other.

Examples:

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Input: N = 5
Output:
{1, 3, 5}
{2, 4}
Explanation: Sum of the subsequence X[] = 1 + 3 + 5 = 9.
Sum of the subsequence Y[] = 2 + 4 = 6.
Since GCD(9, 6) is 3, the sums are not co-prime to each other.

Input: N = 4
Output:
{1, 4}
{2, 3}

Naive Approach: The simplest approach is to split first N natural numbers into two subsequences in all possible ways and for each combination, check if the sum of both the subsequences is non-coprime or not. IF found to be true for any pair of subsequences, print that subsequence and break out of loop

Time Complexity: O(2N)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized based on the following observations:

From the above observation insert all the numbers from the range [1, N] in the one subsequence and N into another subsequence.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// Function to split 1 to N``// into two subsequences``// with non-coprime sums``void` `printSubsequence(``int` `N)``{``    ``cout << ``"{ "``;``    ``for` `(``int` `i = 1; i < N - 1; i++) {``        ``cout << i << ``", "``;``    ``}` `    ``cout << N - 1 << ``" }\n"``;` `    ``cout << ``"{ "` `<< N << ``" }"``;``}``// Driver Code``int` `main()``{``    ``int` `N = 8;` `    ``printSubsequence(N);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``class` `GFG``{``  ` `  ``// Function to split 1 to N``  ``// into two subsequences``  ``// with non-coprime sums``  ``public` `static` `void` `printSubsequence(``int` `N)``  ``{``    ``System.out.print(``"{ "``);``    ``for` `(``int` `i = ``1``; i < N - ``1``; i++)``    ``{``      ``System.out.print(i + ``", "``);``    ``}` `    ``System.out.println(N - ``1` `+ ``" }"``);``    ``System.out.print(``"{ "` `+ N + ``" }"``);``  ``}` `  ``// Driver code``  ``public` `static` `void` `main(String[] args)``  ``{``    ``int` `N = ``8``;``    ``printSubsequence(N);``  ``}``}` `// This code is contributed by divyesh072019`

## Python3

 `# Python3 program for the above approach` `# Function to split 1 to N``# into two subsequences``# with non-coprime sums``def` `printSubsequence(N):``    ` `    ``print``(``"{ "``, end ``=` `"")``    ``for` `i ``in` `range``(``1``, N ``-` `1``):``        ``print``(i, end ``=` `", "``)` `    ``print``(N ``-` `1``, end ``=` `" }\n"``)` `    ``print``(``"{"``, N, ``"}"``)` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``N ``=` `8` `    ``printSubsequence(N)``    ` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program for the above approach``using` `System;``class` `GFG``{``  ` `  ``// Function to split 1 to N``  ``// into two subsequences``  ``// with non-coprime sums``  ``public` `static` `void` `printSubsequence(``int` `N)``  ``{``    ``Console.Write(``"{ "``);``    ``for` `(``int` `i = 1; i < N - 1; i++)``    ``{``        ``Console.Write(i + ``", "``);``    ``}` `    ``Console.WriteLine(N - 1 + ``" }"``);``    ``Console.Write(``"{ "` `+ N + ``" }"``);``  ``}` `  ``// Driver code``  ``public` `static` `void` `Main(``string``[] args)``  ``{``    ``int` `N = 8;``    ``printSubsequence(N);``  ``}``}` `// This code is contributed by AnkThon`

## Javascript

 ``
Output:
```{ 1, 2, 3, 4, 5, 6, 7 }
{ 8 }```

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

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