Given some cards each assigned a number, you're required to select EXACTLY K cards among them.
While you select a card, I will check the number assigned to it and see if it satisfies some of the following conditions:
1. the number is a prime number;
2. the amount of its divisors is a prime number;
3. the sum of its divisors is a prime number;
4. the product of all its divisors is a perfect square number. A perfect square number is such a kind of number that it can be written as a square of an integer.
The score you get from this card is equal to the amount of conditions that its number satisfies. The total score you get from the selection of K cards is equal to the sum of scores of each card you select.
After you have selected K cards, I will check if there's any condition that has never been satisfied by any card you select. If there is, I will add some extra scores to you for each unsatisfied condition. To make the game more interesting, this score may be negative.
After this, you will get your final score. Your task is to figure out the score of each card and find some way to maximize your final score.
Note that 1 is not a prime number. In this problem, we consider a number to be a divisor of itself. For example, considering the number 16, it is not a prime. All its divisors are respectively 1, 2, 4, 8 and 16, and thus, it has 5 divisors with a sum of 31 and a product of 1024. Therefore, it satisfies the condition 2, 3 and 4, which deserves 3 points.