# leetcode\_507

We define the Perfect Number is a positive integer that is equal to the sum of all its positive divisors except itself.

Now, given an integer n, write a function that returns true when it is a perfect number and false when it is not. Example: Input: 28 Output: True Explanation: 28 = 1 + 2 + 4 + 7 + 14 Note: The input number n will not exceed 100,000,000. (1e8)

## Solutions

1. **straight forward O(n^1/2)**

```cpp
class Solution {
public:
    bool checkPerfectNumber(int num) {
        if (num <= 1) return false;
        int sum = 1, i = 2;
        for (; i * i <= num; i++) {
            if (num % i == 0)
                sum += i + num / i;
        }
        if (i * i == num) sum -= i;
        return sum == num;
    }
};
```

1. **math**
2. perfect numbers can be represented as `2^(p-1) * (2^p - 1)` with p is prime number.