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






                        filter_none









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







                        filter_none









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







                        filter_none









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







                        filter_none











Output:


Yes





Time Complexity :  O(log(n))



          
          
          
          

            

    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:
meet97_patel
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