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Number of solutions of n = x + n ⊕ x

  • Last Updated : 22 Mar, 2021

Given a number n, we have to find the number of possible values of X such that n = x + n ⊕ x. Here ⊕ represents XOR

Examples:  

Input : n = 3
Output : 4
The possible values of x are 0, 1, 2, and 3.

Input : n = 2
Output : 2
The possible values of x are 0 and 2.

Brute force approach: We can see that x is always equal to or less than n, so we can iterate over the range [0, n] and count the number of values that satisfy the required condition. The time complexity of this approach is O(n).

C++




// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the number of
// solutions of n = n xor x
int numberOfSolutions(int n)
{
    // Counter to store the number
    // of solutions found
    int c = 0;
 
    for (int x = 0; x <= n; ++x)
        if (n == x + n ^ x)
            ++c;
 
    return c;
}
 
// Driver code
int main()
{
    int n = 3;
    cout << numberOfSolutions(n);
    return 0;
}

Java




// Java implementation of above approach
import java.util.*;
import java.lang.*;
 
class GFG
{
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Counter to store the number
    // of solutions found
    int c = 0;
 
    for (int x = 0; x <= n; ++x)
        if (n == x + (n ^ x))
            ++c;
 
    return c;
}
 
// Driver code
public static void main(String args[])
{
    int n = 3;
    System.out.print(numberOfSolutions(n));
}
}
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

Python3




# Python 3 implementation
# of above approach
 
# Function to find the number of
# solutions of n = n xor x
def numberOfSolutions(n):
 
    # Counter to store the number
    # of solutions found
    c = 0
 
    for x in range(n + 1):
        if (n ==( x +( n ^ x))):
            c += 1
 
    return c
 
# Driver code
if __name__ == "__main__":
    n = 3
    print(numberOfSolutions(n))
 
# This code is contributed
# by ChitraNayal

C#




// C# implementation of above approach
using System;
 
class GFG
{
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Counter to store the number
    // of solutions found
    int c = 0;
 
    for (int x = 0; x <= n; ++x)
        if (n == x + (n ^ x))
            ++c;
 
    return c;
}
 
// Driver code
public static void Main()
{
    int n = 3;
    Console.Write(numberOfSolutions(n));
}
}
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

PHP




<?php
// PHP implementation of above approach
 
// Function to find the number of
// solutions of n = n xor x
function numberOfSolutions($n)
{
    // Counter to store the number
    // of solutions found
    $c = 0;
 
    for ($x = 0; $x <= $n; ++$x)
        if ($n == $x + $n ^ $x)
            ++$c;
 
    return $c;
}
 
// Driver code
$n = 3;
echo numberOfSolutions($n);
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

Javascript




<script>
 
    // Javascript implementation of above approach
     
    // Function to find the number of
    // solutions of n = n xor x
    function numberOfSolutions(n)
    {
         
        // Counter to store the number
        // of solutions found
        let c = 0;
 
        for(let x = 0; x <= n; ++x)
            if (n == x + n ^ x)
                ++c;
 
        return c;
    }
     
    let n = 3;
    document.write(numberOfSolutions(n));
     
    // This code is contributed by divyesh072019.
</script>
Output: 
4

 

Time complexity: O(n)



Efficient approach: We can solve this problem in a more efficient way if we consider n in its binary form. If a bit of n is set, i.e. 1, then we can deduce that there must be a corresponding set bit in either x or n ⊕ x (but not both). If the corresponding bit is set in x, then it is not set in n ⊕ x as 1 ⊕ 1 = 0. Otherwise the bit is set in n ⊕ x as 0 ⊕ 1 = 1. Therefore for every set bit in n, we can have either a set bit or an unset bit in x. However, we cannot have a set bit in x corresponding to an unset bit in n. By this logic, the number of solutions comes out to be 2 raised to the power of the number of set bits in n. The time complexity of this approach is O(log n). 

C++




// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the number of
// solutions of n = n xor x
int numberOfSolutions(int n)
{
    // Number of set bits in n
    int c = 0;
 
    while (n) {
        c += n % 2;
        n /= 2;
    }
 
    // We can also write (1 << c)
    return pow(2, c);
}
 
// Driver code
int main()
{
    int n = 3;
    cout << numberOfSolutions(n);
    return 0;
}

Java




// Java  implementation of above approach
import java.io.*;
 
class GFG {
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Number of set bits in n
    int c = 0;
 
    while (n>0) {
        c += n % 2;
        n /= 2;
    }
 
    // We can also write (1 << c)
    return (int)Math.pow(2, c);
}
 
// Driver code
 
    public static void main (String[] args) {
        int n = 3;
    System.out.println( numberOfSolutions(n));
    }
}
//This code is contributed by anuj_67

Python3




# Python3 implementation of above approach
 
# from math lib. import everything
from math import *
 
# Function to find the number of
# solutions of n = n xor x
def numberOfSolutions(n) :
 
    # Number of set bits in n
    c = 0
 
    while(n) :
        c += n % 2
        n //= 2
 
    # We can also write (1 << c)
    return int(pow(2, c))
 
         
# Driver code    
if __name__ == "__main__" :
 
    n = 3
    print(numberOfSolutions(n))
 
# This code is contributed by ANKITRAI1

C#




// C# implementation of above approach
using System;
 
class GFG
{
// Function to find the number of
// solutions of n = n xor x
static int numberOfSolutions(int n)
{
    // Number of set bits in n
    int c = 0;
 
    while (n > 0)
    {
        c += n % 2;
        n /= 2;
    }
 
    // We can also write (1 << c)
    return (int)Math.Pow(2, c);
}
 
// Driver code
public static void Main ()
{
    int n = 3;
    Console.WriteLine(numberOfSolutions(n));
}
}
 
// This code is contributed by anuj_67

PHP




<?php
// PHP implementation of above approach
 
// Function to find the number of
// solutions of n = n xor x
function numberOfSolutions($n)
{
    // Number of set bits in n
    $c = 0;
    while ($n)
    {
        $c += $n % 2;
        $n /= 2;
    }
 
    // We can also write (1 << c)
    return pow(2, $c);
}
 
// Driver code
$n = 3;
echo numberOfSolutions($n);
 
// This code is contributed by jit_t
?>

Javascript




<script>
 
    // Javascript implementation of above approach
 
    // Function to find the number of
    // solutions of n = n xor x
    function numberOfSolutions(n)
    {
        // Number of set bits in n
        let c = 0;
 
        while (n > 0) {
            c += n % 2;
            n = parseInt(n / 2, 10);
        }
 
        // We can also write (1 << c)
        return Math.pow(2, c);
    }
 
    let n = 3;
    document.write(numberOfSolutions(n));
</script>
Output: 
4

 

Time complexity: O(log n)
 

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