Question: Let $n \in \mathbb{N}$. If $2n$ has $30$ positive factors and $3n$ has $32$ positive factors, how many positive factors does $6n$ have?

the answer is given: <https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_29>

What I have tried so far: I searched for similar question online and I can't understand what the result is saying, here's some of my question: 

(1) what's the relationship between the number of factors and prime factor? Is it just because the prime factor cannot further decompose? 

(2) Is there a way to do this question without testing all possible outcome?

(3) What are the restrictions of this method? Is it applicable in any situation? 

I am just curious when I saw this math competition problem online, and it really tells me I know nothing about number theory... I am in chaos now.