写c++代码，证明一个正整数num，除了1和自身这两个因子之外，如果还有其他因子，那么这个因子最多是sqrt(num),也就是num这个数的平方根。

#include <iostream>
#include <cmath>

using namespace std;

bool is_prime(int num) {
    if (num < 2) return false;
    int limit = sqrt(num);
    for (int i = 2; i <= limit; i++) {
        if (num % i == 0) return false;
    }
    return true;
}

int main() {
    int num;
    cout << "Enter a positive integer: ";
    cin >> num;
    bool has_other_factor = false;
    int limit = sqrt(num);
    for (int i = 2; i <= limit; i++) {
        if (num % i == 0) {
            has_other_factor = true;
            if (is_prime(i)) {
                cout << i << " is a prime factor of " << num << endl;
            } else {
                cout << i << " is a composite factor of " << num << endl;
            }
        }
    }
    if (!has_other_factor) {
        cout << num << " is a prime number." << endl;
    } else {
        cout << "All other factors of " << num << " are not greater than " << limit << endl;
    }
    return 0;
}

输入一个正整数num，程序会检查num是否为质数，如果不是，会输出所有num的因子，其中质因子用“is a prime factor”输出，合数因子用“is a composite factor”输出，并且会判断所有因子是否都小于等于sqrt(num)。