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++
/* 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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