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

Question

Set X consists of exactly four distinct integers that are greater than 1. For each integer in the set, all of that integer’s unique prime factors are also in the set. For example, if 20 is in set X, then 2 and 5 must also be in set X. How many of the distinct integers in set X are prime?

(1) The product of the integers in set X is divisible by 36.

(2) The product of the integers in set X is divisible by 60.

(1) The product of the integers in set X is divisible by 36.

(2) The product of the integers in set X is divisible by 60.

minustark · Answer

1) Case 1: The 4 distinct integers are 36, 2, 3, 6. Then there are 2 primes in the set.
Case 2 : The 4 distinct integers are 6, 2, 3, 7. Then the no of primes is 3. 
Not sufficient.

2) Case 1:The set has 12, 2, 3, 5 as elements. No of primes 3.
Case 2: The set has : 10, 2, 5, 3. Also 3 primes. 
Case 3: The set has 60, 2, 3, 4. 3 primes.
Case 4 : The set has 30, 3, 2, 5....still 3 primes
Sufficient as 60 has 3 primes in it.

B is the answer.

Shiv0817 · Answer

Ans E

St 1 : 2.3 6, 36 - 2 primes
 2,3,5,36 - 3 primes
St 2: 2, 3 , 10 , 5 - 3 primes
 2, 3, 6, 10 - - 2 primes

1+2 - both of St 2 becomes valid hence Not Sufficent

RaghuveerTR · Answer

My understanding is as below.

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

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.

Usernamevisible · Answer

Question Logic
Set X has exactly 4 distinct integers.
Rule:
If a composite number is in X, all of its distinct prime factors must also be in X.
Example:

* If 20 is in X → 2 and 5 must be in X.
* If 12 is in X → 2 and 3 must be in X.
 We need to determine how many members of X are prime.

---

Statement (1)
Product of integers in X is divisible by 36.
36 = (2^2 × 3^2)
Therefore 2 and 3 must appear as prime factors somewhere.
Because all prime factors must themselves belong to X:
2 ∈ X and 3 ∈ X.
But is there a third prime?
Possible Set A:
{2, 3, 4, 9}

* 4's prime factor = 2 ✓
* 9's prime factor = 3 ✓
* Product divisible by 36
 Prime count = 2
 Possible Set B:
 {2, 3, 5, 30}
* 30's prime factors = 2,3,5 ✓
* Product divisible by 36
 Prime count = 3
 Different answers.
 Statement (1) is NOT sufficient.

---

Statement (2)
Product of integers in X is divisible by 60.
60 = (2^2 × 3 × 5)
Therefore 2, 3, and 5 must all appear as prime factors somewhere.
By the rule:
2 ∈ X
3 ∈ X
5 ∈ X
Three of the four elements are already forced.
Now there is only one slot left.
Could that fourth element introduce a new prime factor, say 7?
No.
If 7 were a prime factor of the fourth element, then 7 itself would also have to be in X.
That would require a fifth element.
Impossible.
Therefore the fourth element can only be built from primes 2, 3, and 5.
Examples:
{2,3,5,6}
{2,3,5,10}
{2,3,5,15}
{2,3,5,30}
In every valid case:
Prime elements = 2, 3, 5
Exactly 3 primes.
Statement (2) is sufficient.
----------------------------

Answer
(B)
Statement (2) alone is sufficient, but Statement (1) alone is not sufficient.
-----------------------------------------------------------------------------

Fast takeaway
When you see:
"All prime factors must also be in the set"
and
"Set has exactly 4 elements"
immediately count how many distinct prime factors are forced.

* Statement (1) forces {2,3} → could be 2 primes or 3 primes.
* Statement (2) forces {2,3,5} → already 3 of the 4 slots are taken by primes, so the number of primes is fixed at 3.

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

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

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