Given two positive integers A and B such that A != B, the task is to find a positive integer X which maximizes the expression (A AND X) * (B AND X).
Example:
Input: A = 9 B = 8
Output: 8
(9 AND 8) * (8 AND 8) = 8 * 8 = 64 (maximum possible)
Input: A = 11 and B = 13
Output: 9
Naive approach: One can run a loop from 1 to max(A, B) and can easily find X which maximizes the given expression.
Efficient approach: It is known that,
(a – b)2 ? 0
which implies (a + b)2 – 4*a*b ? 0
which implies a * b ? (a + b)2 / 4
Hence, it concludes that a * b will be maximum when a * b = (a + b)2 / 4
which implies a = b
From the above result, (A AND X) * (B AND X) will be maximum when (A AND X) = (B AND X)
Now X can be found as:
A = 11 = 1011
B = 13 = 1101
X = ? = abcd
At 0th place: (1 AND d) = (1 AND d) implies d = 0, 1 but to maximize (A AND X) * (B AND X) d = 1
At 1st place: (1 AND d) = (0 AND d) implies c = 0
At 2nd place: (0 AND d) = (1 AND d) implies b = 0
At 3rd place: (1 AND d) = (1 AND d) implies a = 0, 1 but to maximize (A AND X) * (B AND X) a = 1
Hence, X = 1001 = 9
Below is the implementation of the above approach:
// C++ implementation of the approach #include <bits/stdc++.h> using namespace std;
#define MAX 32 // Function to find X according // to the given conditions int findX( int A, int B)
{ int X = 0;
// int can have 32 bits
for ( int bit = 0; bit < MAX; bit++) {
// Temporary ith bit
int tempBit = 1 << bit;
// Compute ith bit of X according to
// given conditions
// Expression below is the direct
// conclusion from the illustration
// we had taken earlier
int bitOfX = A & B & tempBit;
// Add the ith bit of X to X
X += bitOfX;
}
return X;
} // Driver code int main()
{ int A = 11, B = 13;
cout << findX(A, B);
return 0;
} |
// C implementation of the approach #include <stdio.h> #define MAX 32 // Function to find X according // to the given conditions int findX( int A, int B)
{ int X = 0;
// int can have 32 bits
for ( int bit = 0; bit < MAX; bit++) {
// Temporary ith bit
int tempBit = 1 << bit;
// Compute ith bit of X according to
// given conditions
// Expression below is the direct
// conclusion from the illustration
// we had taken earlier
int bitOfX = A & B & tempBit;
// Add the ith bit of X to X
X += bitOfX;
}
return X;
} // Driver code int main()
{ int A = 11, B = 13;
printf ( "%d" , findX(A, B));
return 0;
} // This code is contributed by phalasi. |
// Java implementation of the approach class GFG
{ static int MAX = 32 ;
// Function to find X according // to the given conditions static int findX( int A, int B)
{ int X = 0 ;
// int can have 32 bits
for ( int bit = 0 ; bit < MAX; bit++)
{
// Temporary ith bit
int tempBit = 1 << bit;
// Compute ith bit of X according to
// given conditions
// Expression below is the direct
// conclusion from the illustration
// we had taken earlier
int bitOfX = A & B & tempBit;
// Add the ith bit of X to X
X += bitOfX;
}
return X;
} // Driver code public static void main(String []args)
{ int A = 11 , B = 13 ;
System.out.println(findX(A, B));
} } // This code is contributed by 29AjayKumar |
# Python3 implementation of the approach MAX = 32
# Function to find X according # to the given conditions def findX(A, B) :
X = 0 ;
# int can have 32 bits
for bit in range ( MAX ) :
# Temporary ith bit
tempBit = 1 << bit;
# Compute ith bit of X according to
# given conditions
# Expression below is the direct
# conclusion from the illustration
# we had taken earlier
bitOfX = A & B & tempBit;
# Add the ith bit of X to X
X + = bitOfX;
return X;
# Driver code if __name__ = = "__main__" :
A = 11 ; B = 13 ;
print (findX(A, B));
# This code is contributed by AnkitRai01 |
// C# implementation of the approach using System;
class GFG
{ static int MAX = 32;
// Function to find X according // to the given conditions static int findX( int A, int B)
{ int X = 0;
// int can have 32 bits
for ( int bit = 0; bit < MAX; bit++)
{
// Temporary ith bit
int tempBit = 1 << bit;
// Compute ith bit of X according to
// given conditions
// Expression below is the direct
// conclusion from the illustration
// we had taken earlier
int bitOfX = A & B & tempBit;
// Add the ith bit of X to X
X += bitOfX;
}
return X;
} // Driver code public static void Main(String []args)
{ int A = 11, B = 13;
Console.WriteLine(findX(A, B));
} } // This code is contributed by 29AjayKumar |
<script> // Javascript implementation of the approach // Function to find X according // to the given conditions function findX( A, B)
{ var X = 0;
var MAX = 32;
// int can have 32 bits
for ( var bit = 0; bit < MAX; bit++)
{
// Temporary ith bit
var tempBit = 1 << bit;
// Compute ith bit of X according to
// given conditions
// Expression below is the direct
// conclusion from the illustration
// we had taken earlier
var bitOfX = A & B & tempBit;
// Add the ith bit of X to X
X += bitOfX;
}
return X;
} // Driver code var A = 11, B = 13;
document.write(findX(A, B));
// This code is contributed by bunnyram19.
</script> |
9
Time Complexity: O(MAX)
Auxiliary Space: O(1)