First N natural can be divided into two sets with given difference and co-prime sums

Given N and M, task is to find whether numbers 1 to N can be divided into two sets such that the absolute difference between the sum of two sets is M and gcd of the sum of two sets is 1 (i.e. Sum of both sets are co-prime).

Prerequisite : GCD in CPP | GCD

Examples :

Input : N = 5 and M = 7
Output : YES
Explanation : as numbers from 1 to 5 can be divided into two sets {1, 2, 3, 5} and {4} such that absolute difference between the sum of both sets is 11 – 4 = 7 which is equal to M and also GCD(11, 4) = 1.

Input : N = 6 and M = 3
Output : NO
Explanation : In this case, Numbers from 1 to 6 can be divided into two sets {1, 2, 4, 5} and {3, 6} such that absolute difference between their sum is 12 – 9 = 3. But, since 12 and 9 are not co-prime as GCD(12, 9) = 3, the answer is ‘NO’.

Approach : Since we have 1 to N numbers, we know that the sum of all the numbers is N * (N + 1) / 2. Let S1 and S2 be two sets such that,
1) sum(S1) + sum(S2) = N * (N + 1) / 2
2) sum(S1) – sum(S2) = M
Solving these two equations will give us the sum of both the sets. If sum(S1) and sum(S2) are integers and they are co-prime (their GCD is 1), then there exists a way to split the numbers into two sets. Otherwise, there is no way to split these N numbers.

Below is the implementation of the solution described above.

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

/* CPP code to determine whether numbers
   1 to N can be divided into two sets
   such that absolute difference between 
   sum of these two sets is M and these
   two sum are co-prime*/
#include <bits/stdc++.h>
using namespace std;
  
// function that returns boolean value
// on the basis of whether it is possible
// to divide 1 to N numbers into two sets
// that satisfy given conditions.
bool isSplittable(int n, int m)
{
    // initializing total sum of 1
    // to n numbers
    int total_sum = (n * (n + 1)) / 2;
  
    // since (1) total_sum = sum_s1 + sum_s2
    // and (2) m = sum_s1 - sum_s2
    // assuming sum_s1 > sum_s2.
    // solving these 2 equations to get
    // sum_s1 and sum_s2
    int sum_s1 = (total_sum + m) / 2;
  
    // total_sum = sum_s1 + sum_s2
    // and therefore
    int sum_s2 = total_sum - sum_s1;
  
    // if total sum is less than the absolute
    // difference then there is no way we
    // can split n numbers into two sets
    // so return false
    if (total_sum < m)
        return false;
  
    // check if these two sums are
    // integers and they add up to
    // total sum and also if their
    // absolute difference is m.
    if (sum_s1 + sum_s2 == total_sum &&
        sum_s1 - sum_s2 == m)
  
        // Now if two sum are co-prime
        // then return true, else return false.
        return (__gcd(sum_s1, sum_s2) == 1);
  
    // if two sums don't add up to total
    // sum or if their absolute difference
    // is not m, then there is no way to
    // split n numbers, hence return false
    return false;
}
  
// Driver code
int main()
{
    int n = 5, m = 7;
  
    // function call to determine answer
    if (isSplittable(n, m))
        cout << "Yes";
    else
        cout << "No";
  
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

/* Java code to determine whether numbers
1 to N can be divided into two sets
such that absolute difference between 
sum of these two sets is M and these
two sum are co-prime*/
class GFG 
{
    static int GCD (int a, int b)
    {
        return b == 0 ? a : GCD(b, a % b);
    }
      
    // function that returns boolean value
    // on the basis of whether it is possible
    // to divide 1 to N numbers into two sets
    // that satisfy given conditions.
    static boolean isSplittable(int n, int m)
    {
          
        // initializing total sum of 1
        // to n numbers
        int total_sum = (n * (n + 1)) / 2;
      
        // since (1) total_sum = sum_s1 + sum_s2
        // and (2) m = sum_s1 - sum_s2
        // assuming sum_s1 > sum_s2.
        // solving these 2 equations to get
        // sum_s1 and sum_s2
        int sum_s1 = (total_sum + m) / 2;
      
        // total_sum = sum_s1 + sum_s2
        // and therefore
        int sum_s2 = total_sum - sum_s1;
      
        // if total sum is less than the absolute
        // difference then there is no way we
        // can split n numbers into two sets
        // so return false
        if (total_sum < m)
            return false;
      
        // check if these two sums are
        // integers and they add up to
        // total sum and also if their
        // absolute difference is m.
        if (sum_s1 + sum_s2 == total_sum &&
                    sum_s1 - sum_s2 == m)
      
            // Now if two sum are co-prime
            // then return true, else return false.
            return (GCD(sum_s1, sum_s2) == 1);
  
        // if two sums don't add up to total
        // sum or if their absolute difference
        // is not m, then there is no way to
        // split n numbers, hence return false
        return false;
    }
      
    // Driver Code
    public static void main(String args[]) 
    {
        int n = 5, m = 7;
  
        // function call to determine answer
        if (isSplittable(n, m))
            System.out.println("Yes");
        else
            System.out.println("No");
          
    }
}
  
// This code is contributed by Sam007

chevron_right


Python3

# Python3 code to determine whether numbers
# 1 to N can be divided into two sets
# such that absolute difference between
# sum of these two sets is M and these
# two sum are co-prime

def __gcd (a, b):

return a if(b == 0) else __gcd(b, a % b);

# function that returns boolean value
# on the basis of whether it is possible
# to divide 1 to N numbers into two sets
# that satisfy given conditions.
def isSplittable(n, m):

# initializing total sum of 1
# to n numbers
total_sum = (int)((n * (n + 1)) / 2);

# since (1) total_sum = sum_s1 + sum_s2
# and (2) m = sum_s1 – sum_s2
# assuming sum_s1 > sum_s2.
# solving these 2 equations to get
# sum_s1 and sum_s2
sum_s1 = int((total_sum + m) / 2);

# total_sum = sum_s1 + sum_s2
# and therefore
sum_s2 = total_sum – sum_s1;

# if total sum is less than the absolute
# difference then there is no way we
# can split n numbers into two sets
# so return false
if (total_sum < m): return False; # check if these two sums are # integers and they add up to # total sum and also if their # absolute difference is m. if (sum_s1 + sum_s2 == total_sum and sum_s1 - sum_s2 == m): # Now if two sum are co-prime # then return true, else return false. return (__gcd(sum_s1, sum_s2) == 1); # if two sums don't add up to total # sum or if their absolute difference # is not m, then there is no way to # split n numbers, hence return false return False; # Driver code n = 5; m = 7; # function call to determine answer if (isSplittable(n, m)): print("Yes"); else: print("No"); # This code is contributed by mits [tabby title="C#"]

filter_none

edit
close

play_arrow

link
brightness_4
code

/* C# code to determine whether numbers
1 to N can be divided into two sets
such that absolute difference between 
sum of these two sets is M and these
two sum are co-prime*/
using System;
  
class GFG {
  
    static int GCD (int a, int b)
    {
        return b == 0 ? a : GCD(b, a % b);
    }
      
    // function that returns boolean value
    // on the basis of whether it is possible
    // to divide 1 to N numbers into two sets
    // that satisfy given conditions.
    static bool isSplittable(int n, int m)
    {
          
        // initializing total sum of 1
        // to n numbers
        int total_sum = (n * (n + 1)) / 2;
      
        // since (1) total_sum = sum_s1 + sum_s2
        // and (2) m = sum_s1 - sum_s2
        // assuming sum_s1 > sum_s2.
        // solving these 2 equations to get
        // sum_s1 and sum_s2
        int sum_s1 = (total_sum + m) / 2;
      
        // total_sum = sum_s1 + sum_s2
        // and therefore
        int sum_s2 = total_sum - sum_s1;
      
        // if total sum is less than the absolute
        // difference then there is no way we
        // can split n numbers into two sets
        // so return false
        if (total_sum < m)
            return false;
      
        // check if these two sums are
        // integers and they add up to
        // total sum and also if their
        // absolute difference is m.
        if (sum_s1 + sum_s2 == total_sum &&
                       sum_s1 - sum_s2 == m)
      
            // Now if two sum are co-prime
            // then return true, else return false.
            return (GCD(sum_s1, sum_s2) == 1);
  
        // if two sums don't add up to total
        // sum or if their absolute difference
        // is not m, then there is no way to
        // split n numbers, hence return false
        return false;
    }
      
    // Driver code
    public static void Main()
    {
        int n = 5, m = 7;
  
        // function call to determine answer
        if (isSplittable(n, m))
            Console.Write("Yes");
        else
            Console.Write("No");
    }
}
  
// This code is contributed by Sam007.

chevron_right


PHP

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
/* PHP code to determine whether numbers
1 to N can be divided into two sets
such that absolute difference between 
sum of these two sets is M and these
two sum are co-prime*/
  
function __gcd ($a, $b)
{
        return $b == 0 ? $a : __gcd($b, $a % $b);
}
  
// function that returns boolean value
// on the basis of whether it is possible
// to divide 1 to N numbers into two sets
// that satisfy given conditions.
function isSplittable($n, $m)
{
    // initializing total sum of 1
    // to n numbers
    $total_sum = (int)(($n * ($n + 1)) / 2);
  
    // since (1) total_sum = sum_s1 + sum_s2
    // and (2) m = sum_s1 - sum_s2
    // assuming sum_s1 > sum_s2.
    // solving these 2 equations to get
    // sum_s1 and sum_s2
    $sum_s1 = (int)(($total_sum + $m) / 2);
  
    // total_sum = sum_s1 + sum_s2
    // and therefore
    $sum_s2 = $total_sum - $sum_s1;
  
    // if total sum is less than the absolute
    // difference then there is no way we
    // can split n numbers into two sets
    // so return false
    if ($total_sum < $m)
        return false;
  
    // check if these two sums are
    // integers and they add up to
    // total sum and also if their
    // absolute difference is m.
    if ($sum_s1 + $sum_s2 == $total_sum &&
        $sum_s1 - $sum_s2 == $m)
  
        // Now if two sum are co-prime
        // then return true, else return false.
        return (__gcd($sum_s1, $sum_s2) == 1);
  
    // if two sums don't add up to total
    // sum or if their absolute difference
    // is not m, then there is no way to
    // split n numbers, hence return false
    return false;
}
  
// Driver code
$n = 5;
$m = 7;
  
// function call to determine answer
if (isSplittable($n, $m))
    echo "Yes";
else
    echo "No";
  
// This Code is Contributed by mits
?>

chevron_right


Output:

Yes

Time Complexity : O(log(n))



My Personal Notes arrow_drop_up

A competitive coder developer and a learner by choice who is always eager to contribute to the computer science and developer community

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Improved By : Sam007, Mithun Kumar