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’.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++

/* 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; 
} 

Java

/* 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 

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 

C#

/* 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. 

PHP

<?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 
?> 

Output:

Yes

Time Complexity : O(log(n))

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Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP

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