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

Question

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

(1) The product of any three integers in the set is negative.
 
(2) The product of the smallest and largest integers in the set is a prime number.

Zarrolou · Accepted Answer

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

(1) The product of any three integers in the list is negative 
Case {-1,1,2} no negative.
Case {-3,-2,-1} all negative.
Not sufficient

(2) The product of the smallest and largest integers in the list is a prime number. 
So their product is positive
Case {1,2,3} no negative.
Case {-3,-2,-1} all negative.
Not sufficient

(1)+(2)  From 2 we get that the smallest and the largest have the same sign. From one we get that the product of any three integers is negative, so every number is negative. 
If just one is positive than the product of -*-*+ would be + and statement one would not hold true.
 C

Bunuel · Answer

Set S consists of more than two integers. Are all the integers in set S negative? 
 
(1) The product of any three integers in the set is negative. 
 
If the set consists of only three terms, then it could be {negative, negative, negative}, giving an YES answer to the question, or {negative, positive, positive}, giving an NO answer to the question. However, if the set consists of more than three terms, then it can only have negative numbers, also giving an YES answer to the question. Not sufficient.
 
(2) The product of the smallest and largest integers in the set is a prime number. 
 
Since only positive numbers can be primes, the smallest and largest integers in the set must be of the same sign. Thus, the set consists of either only negative or only positive integers. Not sufficient.
 
(1)+(2) Statement (2) tells us that the set consists of either only negative or only positive integers. This eliminates the possibility of the set having the form {negative, positive, positive} from statement (1), which was the only case in (1) giving a NO answer. Therefore, in statement (1), we are left with the sets consisting of only negative integers. Sufficient.

Answer: C

amanvermagmat · Answer

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 .

_shashank_shekhar_ · 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.

pushpitkc · Answer

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!

_shashank_shekhar_ · Answer

Oh yess... I dont know what i was thinking. Thanks a lot.. got it now.

Sent from my Moto G (5S) Plus using GMAT Club Forum mobile app

BrentGMATPrepNow · Answer

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

briantoth6 · Answer

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.

exc4libur · Answer

set {n > 2 integers}

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

(2) The product of the smallest and largest integers in the list is a prime number; insufic.
  (smallest*largest)=-13*-1=13=prime and the set: {-13…-1} all neg
  (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)  is not valid, thus all are negative, sufic.

Kristians4 · Answer

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

BrentGMATPrepNow · Answer

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.

CMBR2022 · Answer

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?

Pritishd · Answer

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

CMBR2022 · Answer

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

apoorvashobhitha · Answer

Statement-I: The product obtained by multiplying any three distinct integers from the series is negative.
If the set consists of only three terms, then it could be {negative, negative, negative}, If the set consists of only three terms, it could be
{negative, negative, negative}, giving a YES answer, or
{negative, positive, positive}, giving a NO answer.
If the set consists of more than three terms, then it can only have negative numbers, giving a YES answer.
Therefore, both YES and NO answers are possible.
Thus, Statement I ALONE is NOT sufficient.
Statement-II: The product of the least and the greatest integers in the series results in a prime number.
Since the product is positive, the least and greatest integers must have the same sign.

Therefore, the set could consist of:
All negative integers, or
All positive integers
Or {negative, positive, negative}, {positive, negative, positive}
Because Middle integers are not restricted by Statement II:
They could be positive or negative, as long as the least and greatest integers have the same sign.
Examples for a three-term set:
{−3, −2, −1} → all negative
{−3, 2, −1} → middle positive, least and greatest negative
{1, 2, 3} → all positive
{1, −2, 3} → middle negative, least and greatest positive
Hence, it cannot be determined whether the integers are all negative or all positive.
Thus, Statement II ALONE is NOT sufficient.
Combining both:
From Statement: II, the least and greatest integers have the same sign and their product is a prime number.
From Statement: I, the product of any three distinct integers from the series is negative.

Consider possible cases:

• If all integers are positive, the product of any three would be positive, which contradicts Statement: I. So all-positive is impossible.

• If all integers are negative, every triple product is negative and Statement: II can hold if the extremes multiply to a prime. This yields answer YES.

• If the extremes are positive and some middle term is negative (for a three-term series, for example {1, −2, 3}), Statement: II holds (1×3 = 3, prime) and Statement: I holds (the only triple product is negative). This yields answer NO.

Both YES and NO are possible under the combined statements
The question cannot be answered even using any of the statements

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

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

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