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

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:**

Input:N = 5Output:

{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 = 4Output:

{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(2^{N})**Auxiliary Space:** O(1)

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

- The sum of the first
**(N – 1)**natural numbers is given by**(N*(N – 1))/2**. - Therefore, GCD of
**((N*(N – 1))/2)**and**N**is**N**.

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 <iostream>` `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

`<script>` ` ` `// Javascript program for the above approach` ` ` ` ` `// Function to split 1 to N` ` ` `// into two subsequences` ` ` `// with non-coprime sums` ` ` `function` `printSubsequence(N)` ` ` `{` ` ` `document.write(` `"{ "` `);` ` ` `for` `(let i = 1; i < N - 1; i++)` ` ` `{` ` ` `document.write(i + ` `", "` `);` ` ` `}` ` ` `document.write(N - 1 + ` `" }"` `+ ` `"</br>"` `);` ` ` `document.write(` `"{ "` `+ N + ` `" }"` `+ ` `"</br>"` `);` ` ` `}` ` ` ` ` `let N = 8;` ` ` `printSubsequence(N);` ` ` `</script>` |

**Output:**

{ 1, 2, 3, 4, 5, 6, 7 } { 8 }

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

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