A better wording for the problem might be as follows:

If k is a positive integer, is k the square of an integer?

1. k+1 is has exactly two different positive divisors.

2. k has exactly 9 different positive divisors.

Statement 1: k+1 is has exactly two different positive divisors.
In other words, k+1 is PRIME.
Case 1: k+1 = 2, in which case k=1
Here, k is a perfect square.
Case 2: k+1 = 3, in which case k=2
Here, k is NOT a perfect square.
INSUFFICIENT.

Statement 2: k has exactly 9 different positive divisors.
Only a perfect square has an odd number of positive divisors.
Thus, k is a perfect square.
SUFFICIENT.

The correct answer is B.

Where I can find summary of such rules.

Only a perfect square has an odd number of positive divisors.

I tried using to find number of factors using multiplying the (powers +1) of prime factors.

e.g 9 = 3^2 so 9 has 3 factors

I tried using to find number of factors using multiplying the (powers +1) of prime factors.

You had the right idea. If you'd taken 2^8, for example, you can see that there would be 8+1 = 9 factors. And 2^8 is the same as (2^4)^2, so it's clearly a perfect square.

Similarly 3^8 will also have 9 factors, and 3^8 = (3^4)^2, so this is also a perfect square. Note: any prime base raised to an even exponent will be a perfect square, because squaring a number is the equivalent of multiplying all of its exponent values by 2.

Put another way, any positive number, when expressed as a product of its prime bases, will be a perfect square if each of those prime bases is raised to an even exponent. So (2^10) * (3^12) is a perfect square, as '10' and '12' are both even. But (2^9 * 3^12) is not a perfect square, as '9' is not even.

The point: rather than memorizing a list of rules, which can offer some benefits, it's often better to really understand the mechanics and logic of those rules, so they can be applied to other scenarios.