# Split N natural numbers into two sets having GCD of their sums greater than 1

Given an integer **N**, the task is to create two sets of distinct elements from **1 to N **such that the gcd of their respective sums is greater than 1. Print the respective sets. If no such split can be done, print -1.

**Examples:**

Input:N = 5Output:

2 4

1 3 5Explanation:

GCD(Sum({2, 4}), Sum({1, 3, 5}) = GCD(6, 9) = 3

Input:N = 2Output:-1Explanation:

For N = 2, it is not possible to divide into two sets having GCD of their sum greater than 1.

**Approach 1:** A simple approach is to split all even numbers up to N into one set and all odd numbers into another set.

Below code is the implementation of the approach:

## C++

`// C++ program to split N natural numbers` `// into two sets having GCD` `// of their sums greater than 1` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to create` `// and print the two sets` `void` `createSets(` `int` `N)` `{` ` ` `// No such split` ` ` `// possible for N <= 2` ` ` `if` `(N <= 2) {` ` ` `cout << ` `"-1"` `<< endl;` ` ` `return` `;` ` ` `}` ` ` `// Print the first set` ` ` `// consisting of even elements` ` ` `for` `(` `int` `i = 2; i <= N; i += 2)` ` ` `cout << i << ` `" "` `;` ` ` `cout << ` `"\n"` `;` ` ` `// Print the second set` ` ` `// consisting of odd ones` ` ` `for` `(` `int` `i = 1; i <= N; i += 2) {` ` ` `cout << i << ` `" "` `;` ` ` `}` `}` `// Driver Code` `int` `main()` `{` ` ` `int` `N = 6;` ` ` `createSets(N);` ` ` `return` `0;` `}` |

## Java

`// Java program to split N natural numbers` `// into two sets having GCD` `// of their sums greater than 1` `class` `GFG{` ` ` `// Function to create` `// and print the two sets` `static` `void` `createSets(` `int` `N)` `{` ` ` `// No such split` ` ` `// possible for N <= 2` ` ` `if` `(N <= ` `2` `)` ` ` `{` ` ` `System.out.println(` `"-1"` `);` ` ` `return` `;` ` ` `}` ` ` `// Print the first set` ` ` `// consisting of even elements` ` ` `for` `(` `int` `i = ` `2` `; i <= N; i += ` `2` `)` ` ` `System.out.print(i + ` `" "` `);` ` ` `System.out.print(` `"\n"` `) ;` ` ` ` ` `// Print the second set` ` ` `// consisting of odd ones` ` ` `for` `(` `int` `i = ` `1` `; i <= N; i += ` `2` `)` ` ` `{` ` ` `System.out.print(i + ` `" "` `);` ` ` `}` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `int` `N = ` `6` `;` ` ` `createSets(N);` `}` `}` `// This code is contributed by shivanisinghss2110` |

## Python3

`# Python3 program to split N natural numbers` `# into two sets having GCD` `# of their sums greater than 1` `# Function to create` `# and print the two sets` `def` `createSets(N):` ` ` ` ` `# No such split` ` ` `# possible for N <= 2` ` ` `if` `(N <` `=` `2` `):` ` ` `print` `(` `"-1"` `);` ` ` `return` `;` ` ` `# Print the first set` ` ` `# consisting of even elements` ` ` `for` `i ` `in` `range` `(` `2` `, N ` `+` `1` `, ` `2` `):` ` ` `print` `(i, end` `=` `" "` `);` ` ` `print` `("");` ` ` `# Print the second set` ` ` `# consisting of odd ones` ` ` `for` `i ` `in` `range` `(` `1` `, N ` `+` `1` `, ` `2` `):` ` ` `print` `(i, end ` `=` `" "` `);` `# Driver Code` `if` `__name__ ` `=` `=` `'__main__'` `:` ` ` `N ` `=` `6` `;` ` ` `createSets(N);` `# This code is contributed by gauravrajput1` |

## C#

`// C# program to split N natural numbers` `// into two sets having GCD` `// of their sums greater than 1` `using` `System;` `class` `GFG{` `// Function to create` `// and print the two sets` `static` `void` `createSets(` `int` `N)` `{` ` ` `// No such split` ` ` `// possible for N <= 2` ` ` `if` `(N <= 2)` ` ` `{` ` ` `Console.WriteLine(` `"-1"` `);` ` ` `return` `;` ` ` `}` ` ` `// Print the first set` ` ` `// consisting of even elements` ` ` `for` `(` `int` `i = 2; i <= N; i += 2)` ` ` `Console.Write(i + ` `" "` `);` ` ` `Console.Write(` `"\n"` `) ;` ` ` `// Print the second set` ` ` `// consisting of odd ones` ` ` `for` `(` `int` `i = 1; i <= N; i += 2)` ` ` `{` ` ` `Console.Write(i + ` `" "` `);` ` ` `}` `}` `// Driver Code` `public` `static` `void` `Main(String[] args)` `{` ` ` `int` `N = 6;` ` ` `createSets(N);` `}` `}` `// This code is contributed by shivanisinghss2110` |

## Javascript

`<script>` ` ` `// Javascript program to split N natural numbers` ` ` `// into two sets having GCD` ` ` `// of their sums greater than 1` ` ` `// Function to create` ` ` `// and print the two sets` ` ` `function` `createSets(N)` ` ` `{` ` ` `// No such split` ` ` `// possible for N <= 2` ` ` `if` `(N <= 2) {` ` ` `document.write(` `"-1"` `);` ` ` `return` `;` ` ` `}` ` ` `// Print the first set` ` ` `// consisting of even elements` ` ` `for` `(let i = 2; i <= N; i += 2)` ` ` `document.write(i + ` `" "` `);` ` ` `document.write(` `"</br>"` `);` ` ` `// Print the second set` ` ` `// consisting of odd ones` ` ` `for` `(let i = 1; i <= N; i += 2) {` ` ` `document.write(i + ` `" "` `);` ` ` `}` ` ` `}` ` ` ` ` `let N = 6;` ` ` `createSets(N);` `</script>` |

**Output:**

2 4 6 1 3 5

Time Complexity: O(N)

Auxiliary Space: O(1),since no extra space has been taken.

**Approach 2:** This approach can be divided into 2 cases based on N:

**If N is odd:**- The sum of N natural numbers is divisible by
**(N+1)/2**in this case. - Therefore we only need to put
**(N+1)/2 into first set**and the remaining numbers into second set. - This will automatically make the GCD of their sums greater than 1.

- The sum of N natural numbers is divisible by
**If N is even:**- The sum of N natural numbers is divisible by
**N/2**in this case. - Therefore we only need to put
**N/2 into first set**and the remaining numbers into second set. - This will automatically make the GCD of their sums greater than 1.

- The sum of N natural numbers is divisible by

Below is the implementation of the above approach:

## C++

`// C++ program to split N natural numbers` `// into two sets having GCD` `// of their sums greater than 1` `#include <bits/stdc++.h>` `using` `namespace` `std;` `#define ll long long` `// Function to create` `// and print the two sets` `void` `createSets(ll n)` `{` ` ` `// For n <= 2 such sets` ` ` `// can never be formed` ` ` `if` `(n <= 2) {` ` ` `cout << ` `"-1"` `;` ` ` `return` `;` ` ` `}` ` ` `else` `{` ` ` `// Check if N is even or odd` ` ` `// and store the element` ` ` `// which divides the sum of N` ` ` `// natural numbers accordingly` ` ` `ll x = (n % 2 == 0) ? (n / 2)` ` ` `: ((n + 1) / 2);` ` ` `// First set` ` ` `cout << x << endl;` ` ` `// Print elements of second set` ` ` `for` `(ll i = 1; i <= n; i++) {` ` ` `if` `(i == x)` ` ` `continue` `;` ` ` `cout << i << ` `" "` `;` ` ` `}` ` ` `}` ` ` `return` `;` `}` `// Driver code` `int` `main()` `{` ` ` `ll N = 7;` ` ` `createSets(N);` ` ` `return` `0;` `}` |

## Java

`// Java program to split N natural numbers` `// into two sets having GCD` `// of their sums greater than 1` `class` `GFG{` `// Function to create` `// and print the two sets` `static` `void` `createSets(` `long` `n)` `{` ` ` ` ` `// For n <= 2 such sets` ` ` `// can never be formed` ` ` `if` `(n <= ` `2` `)` ` ` `{` ` ` `System.out.print(` `"-1"` `);` ` ` `return` `;` ` ` `}` ` ` `else` ` ` `{` ` ` ` ` `// Check if N is even or odd` ` ` `// and store the element` ` ` `// which divides the sum of N` ` ` `// natural numbers accordingly` ` ` `long` `x = (n % ` `2` `== ` `0` `) ? (n / ` `2` `) :` ` ` `((n + ` `1` `) / ` `2` `);` ` ` `// First set` ` ` `System.out.print(x + ` `"\n"` `);` ` ` `// Print elements of second set` ` ` `for` `(` `int` `i = ` `1` `; i <= n; i++)` ` ` `{` ` ` `if` `(i == x)` ` ` `continue` `;` ` ` `System.out.print(i + ` `" "` `);` ` ` `}` ` ` `}` ` ` `return` `;` `}` `// Driver code` `public` `static` `void` `main(String[] args)` `{` ` ` `long` `N = ` `7` `;` ` ` `createSets(N);` `}` `}` `// This code is contributed by PrinciRaj1992` |

## Python3

`# Python3 program to split N natural numbers` `# into two sets having GCD` `# of their sums greater than 1` `# Function to create` `# and print the two sets` `def` `createSets(n):` ` ` ` ` `# For n <= 2 such sets` ` ` `# can never be formed` ` ` `if` `(n <` `=` `2` `):` ` ` `print` `(` `"-1"` `);` ` ` `return` `;` ` ` `else` `:` ` ` `# Check if N is even or odd` ` ` `# and store the element` ` ` `# which divides the sum of N` ` ` `# natural numbers accordingly` ` ` `x ` `=` `(n ` `/` `/` `2` `) ` `if` `(n ` `%` `2` `=` `=` `0` `) ` `else` `(` ` ` `(n ` `+` `1` `) ` `/` `/` `2` `);` ` ` `# First set` ` ` `print` `(x, ` `" "` `);` ` ` `# Print elements of second set` ` ` `for` `i ` `in` `range` `(` `1` `, n ` `+` `1` `):` ` ` `if` `(i ` `=` `=` `x):` ` ` `continue` `;` ` ` `print` `(i, end ` `=` `" "` `);` ` ` `return` `;` `# Driver code` `if` `__name__ ` `=` `=` `'__main__'` `:` ` ` ` ` `N ` `=` `7` `;` ` ` `createSets(N);` `# This code is contributed by 29AjayKumar` |

## C#

`// C# program to split N natural numbers` `// into two sets having GCD` `// of their sums greater than 1` `using` `System;` `class` `GFG{` `// Function to create` `// and print the two sets` `static` `void` `createSets(` `long` `n)` `{` ` ` ` ` `// For n <= 2 such sets` ` ` `// can never be formed` ` ` `if` `(n <= 2)` ` ` `{` ` ` `Console.Write(` `"-1"` `);` ` ` `return` `;` ` ` `}` ` ` `else` ` ` `{` ` ` ` ` `// Check if N is even or odd` ` ` `// and store the element` ` ` `// which divides the sum of N` ` ` `// natural numbers accordingly` ` ` `long` `x = (n % 2 == 0) ? (n / 2) :` ` ` `((n + 1) / 2);` ` ` `// First set` ` ` `Console.Write(x + ` `"\n"` `);` ` ` `// Print elements of second set` ` ` `for` `(` `int` `i = 1; i <= n; i++)` ` ` `{` ` ` `if` `(i == x)` ` ` `continue` `;` ` ` `Console.Write(i + ` `" "` `);` ` ` `}` ` ` `}` ` ` `return` `;` `}` `// Driver code` `public` `static` `void` `Main(String[] args)` `{` ` ` `long` `N = 7;` ` ` `createSets(N);` `}` `}` `// This code is contributed by sapnasingh4991` |

## Javascript

`<script>` ` ` `// Javascript program to split N natural numbers` ` ` `// into two sets having GCD` ` ` `// of their sums greater than 1` ` ` ` ` `// Function to create` ` ` `// and print the two sets` ` ` `function` `createSets(n)` ` ` `{` ` ` ` ` `// For n <= 2 such sets` ` ` `// can never be formed` ` ` `if` `(n <= 2) {` ` ` `document.write(` `"-1"` `);` ` ` `return` `;` ` ` `}` ` ` `else` `{` ` ` `// Check if N is even or odd` ` ` `// and store the element` ` ` `// which divides the sum of N` ` ` `// natural numbers accordingly` ` ` `let x = (n % 2 == 0) ? (n / 2) : ((n + 1) / 2);` ` ` `// First set` ` ` `document.write(x + ` `"</br>"` `);` ` ` `// Print elements of second set` ` ` `for` `(let i = 1; i <= n; i++)` ` ` `{` ` ` `if` `(i == x)` ` ` `continue` `;` ` ` `document.write(i + ` `" "` `);` ` ` `}` ` ` `}` ` ` `return` `;` ` ` `}` `// Driver code` ` ` `let N = 7;` ` ` `createSets(N);` ` ` ` ` `// This code is contributed by suresh07.` `</script>` |

**Output:**

4 1 2 3 5 6 7

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