Check if a number is a power of 2 with a single bitwise operation
Given a positive integer
n, you are asked to find out if it’s a power of 2. It’s possible to do that with a single bitwise operation. If we consider the bitwise representation of a number that’s a power of 2, then it has a single 1 in the representation and the rest are 0s. number | Binary representation |
8 | 1000 |
32 | 100000 |
128 | 10000000 |
Random Number | Binary representation |
7 | 111 |
67 | 1000011 |
144 | 10010000 |
This can be seen from the definition of binary representation. Every position with a 1 corresponds to the addition of that power of 2 (67 = 1 + 2 + 0 + 0 + 0 + 0 + 64, 8 = 0 + 0 + 0 + 8). So, having a single
1 in the binary representation of a number means that there are no other powers of 2 in the sum other than that one. So, the problem becomes checking if there is only a single 1 in the binary representation of n.An easy way of doing that is by doing a bitwise
& of n and n-1. If the result is 0 ⇒ it’s a power of 2. For any number that’s not a power of 2, the result of
n & (n-1) will be different from 0:- 7 & 6 = 111 & 110 = 110 (6)
- 67 & 66 = 1000011 & 1000010 = 1000010 (66)
- 144 & 143 = 10010000 & 10001111 = 10000000 (128)
Yet, for the powers of 2, it’s always 0:
- 8 & 7 = 1000 & 111 = 0
- 32 & 31 = 100000 & 11111 = 0
- 128 & 127 = 10000000 & 1111111 = 0
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Intuition
When considering the numbers
n and n-1, the most significant bit of those two is preserved only when the binary representation of n has other 1s other than the most significant bit. If n is a power of 2 ⇒ the binary representation has only a single 1, and thus when subtracting 1, n-1 loses the most significant bit (it becomes 0 and the rest of the bits turn to 1).
For other numbers (6, 67, 144, etc.) their binary representations contain several 1s. Therefore, when subtracting 1 from them, they don’t lose their most significant bits.Input
The first line of the input contains a single integer
n (2 ≤ n ≤ ).Output
The program should print
Yes if n is a power of 2 and No otherwise.Examples
Input | Output |
8 | Yes |
17 | No |
2048 | Yes |
Constraints
Time limit: 2 seconds
Memory limit: 512 MB
Output limit: 1 MB