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Set X consists of exactly four distinct integers that are greater than 29 Oct 2020, 12:02
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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.
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B
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Set X consists of exactly four distinct integers that are greater than 29 Oct 2020, 12:18
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Sajjad1994
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.
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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.
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Set X consists of exactly four distinct integers that are greater than 31 Oct 2020, 00:07
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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
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Set X consists of exactly four distinct integers that are greater than 01 Nov 2020, 09:40
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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.
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Set X consists of exactly four distinct integers that are greater than 17 Apr 2026, 00:52
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