Set X consists of exactly four distinct integers that are greater than

My understanding is as below.

[St 1] product of the integers is divisible by 36

The set could be 36, 2, 3, 9. Number of primes will be 2. (or)
The set could be 180, 2, 3, 5. Number of primes will be 3. Since 180 is a multiple of 36, the product of all integers in the set would be a multiple of 36. Also, 5 is a prime factor of 180 other than 2 and 3, so it must be present in the set.

So, insufficient

[St 2] product of the integers is divisible by 60

The set could be 60, 2, 3, 5. Number of primes will be 3. (or)
The set could be 60*2, 2, 3, 5. Number of primes will still be 3.

Here, I first thought, if 60*13 is in the set, then the number of primes will change. But, this will tear apart the conditions in the question stem: number of elements is 4 and all the prime factors of a number must be present.

So, 60*(any prime number other than 2, 3, 5) cannot be in the set as 60 itself has contributed for 3 prime factors already: 2, 3, 5.

Ultimately, the set will be of the form X = {2, 3, 5, 2n} where n is a integer that has no prime factors other than 2, 3 and 5

In all the cases, the number of prime integers in the set will be 3. Sufficient.

Why are we considering the product to be a part of the set? Did not get that part Bunuel please help.