Set S consists of more than two integers. Are all the integers in set

So Set S can have three or more integers.

(1) Product of any three integers is negative. Now if set S has only three integers, then the product of 'any three integers' means product of these three integers only. And if product of these three integers is negative, it could mean either that all three are negative Or only one of them is negative. So this is giving us two different possibilities, and so Insufficient.
If instead set has more than three integers (say four or five or hundred whatever), then for product of 'any three integers' to be negative it would only be possible if ALL the integers in set are negative.
So from this statement, we get that if set S has exactly three integers - then either all are negative or only one is negative. But if set S has more than three integers, then all must be negative. Not Sufficient.

(2) Prime numbers are always positive. So if we arrange the integers of set S in say ascending order, the product of smallest (first integer) and largest (last integer) is coming out to be positive. This can happen if both smallest and largest integers are negative (in which case all integers will be negative) Or if both smallest and largest integers are positive (in which case all integers will be positive).
So from this statement, we get that either all integers in set S are negative or all integers in set S are positive. Not Sufficient.

Combining both, whether it be a set with exactly three integers or more than three integers - from second statement we know that either all integers in set S are negative or all integers in set S are positive. But if all integers become positive, then product of 'any three integers' will NEVER be negative, it will be positive and will violate the first statement. So we are left with only one case - that all integers in set S are negative. Sufficient.

Hence C answer.

I have a silly question.
Why case (+, - ,+) is not considered here?
For ex- from 1 and 2
Apart from (-, - , -) which is discussed above, If we take (1,-4, 3) as an example, then also we are getting the desired result.
So I marked the answer as E.
Plz someone clarify my doubt.

Hi _shashank_shekhar_

The case you have taken is not possible because the product of the smallest
and largest number must be a prime number as per statement 2.
In the example you have taken, the product will be -12 which is not prime.

Hope that helps you!

Oh yess... I dont know what i was thinking.

Thanks a lot.. got it now.

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Target question: Are all the numbers in set S negative?

Statement 1: The product of any three integers in the list is negative
There are only 2 scenarios in which the product of 3 integers is negative.
scenario #1: all 3 integers are negative
scenario #2: 2 integers are positive, and 1 integer is negative

So, here are two possible cases that satisfy statement 1:
Case a: set S = {-3, -2, -1}, in which case all of the numbers in set S are negative
Case b: set S = {-1, 1, 3}, in which case not all of the numbers in set S are negative
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The product of the smallest and largest integers in the list is a prime number.
Here are two possible cases that satisfy statement 2:
Case a: set S = {-3, -2, -1}, in which case all of the numbers in set S are negative
Case b: set S = {1, 2, 3}, in which case not all of the numbers in set S are negative
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Earlier, we learned that, if the product of 3 integers is negative, then there are 2 possible scenarios:
- scenario #1: all 3 integers are negative
- scenario #2: 2 integers are positive, and 1 integer is negative

Statement 2 tells us that the product of the smallest and largest integers in the list is a prime number. In other word, the product of the smallest and largest integers is POSITIVE.
This allows us to eliminate scenario #2, because under this scenario, the smallest integer in set S would be negative and the largest would be positive, so the product would be NEGATIVE (and prime numbers, by definition, are positive)

This leaves us with scenario #1.
From here, we can conclude that, if the product of any three integers is ALWAYS negative, then ALL of the integers in the set must be negative.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

but it doesn't specify that all the #'s in S must be integers.

what about set {-7, -3, -1, .2}

This technically complies with both statements (1) and (2), a bit misleading given that the question asks about whether all numbers are negative.

set {n > 2 integers}

(1) The product of any three integers in the list is negative: insufic.
[A] {-1,-1,-1}=-1 all neg
[B] {-1,2,3}=-6 not all neg

(2) The product of the smallest and largest integers in the list is a prime number; insufic.
[A] (smallest*largest)=-13*-1=13=prime and the set: {-13…-1} all neg
[B] (smallest*largest)=1*5=5=prime and the set: {1…5} not all neg

(1&2) numbers in set must have same sign, so case (1)[B] is not valid, thus all are negative, sufic.

Hi Brent, does it not matter if there's 4+ numbers in the set?

There could be four or more numbers in the set, but for the purposes of determining the sufficiency of statement 1, we don't need to deal with that.
All we need to show is that there are possible cases in which we get different answers to the target question, at which point we can conclude that statement 2 is not sufficient.

How this is reasoning complete? You've missed the scenario (1, -2, 3)
I- This satisfies case 1- Product is negative
II- Satisfies case 2- 1*3 is prime

BUT these numbers are NOT negative. The answer cannot be C. It should be 'E'. Please explain why one shouldn't consider this case?

Nope!

The second statement states that the product of the SMALLEST and the LARGEST integer in the set must be a prime. In your example {1, -2, 3} the smallest integer is -2 and not 1 and -2 * 3 = -6 which is neither a prime nor positive

I guess the order in which you have put together the set confused you into assuming that 1 is the smallest integer

The answer C stands true

I see.. yeah, I did that mistake for sure- thanks for the clarification and the catch!

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