M28-53

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.

Bunuel · Accepted Answer

Official Solution:

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

King407 · Answer

Hi,

It seems you are trying to falsify what is already established in the statement. Statement A states that product of ANY three nos. is negative. Hence we take only those sets of nos. where this condition holds true.

Statement A states that the product of ANY 3 nos. in the set is negative. This can only be true when all the nos. are negative or one or two nos. are positive based on the size of the set. (if it is a three no. set, then two positive nos., for more than 3 no. sets, 1 positive no.) {simply because if there are 3 positive nos., it will hold the Statement A false}. As nos. can be either positive(one or two) or negative, Hence A is not sufficient.

And B states that the product of min and max value is prime. Since only positive nos. can be prime, min and max values should both be either positive or negative. Again B is not sufficient

Thus taking A & C together, both min and max can not be positive since A has already established that all the nos. can not be positive (if min and max are both positive this means all the nos., lie between min and max, will also be positive)
Thus if min and max are both negative, all the nos. are negative which back the inference we made in A.

Hence C is the answer.

aimtoteach · Answer

Hello! I didnt get this one. What if the set has 4 values - 2 positive and 2 negative. -> Now if we pick any 3 values it can be either positive or negative. e.g 2*3*-1 = -6 but -5*-1*2 - 10 Can you please explain. :oops:

Thanks!

appu5 · Answer

I didn't get it how is it C ?

and the order can be { positive, negative , positive } rite then how is c correct ?

Naina1 · Answer

Hi appu5
{ positive, negative , positive }
In this case the first condition holds true i.e. product of 3 no's is negative.
Now, lets check he 2nd condition. The product of smallest and largest no. is a prime no.Here smallest would be the negative no. and largest would be one of the positive no., that gives us a negative no.(prime no's are always postiive). So, this example does not satisfy the 2nd condition.

harshada1 · Answer

I think this is a  poor-quality  question and I don't agree with the explanation. Set (1,-2,3 ) and (-1,-2,-3) can satisfy both conditions

Bunuel · Answer

You did not understand the question and the solution.

(2) says: the product of the  smallest  and  largest  integers in the set is a prime number.

The product of the smallest and largest integers of (1, -2, 3) is -2*3 = -6, which is not a prime.

njj1984 · Answer

I think this is a  poor-quality  question and I don't agree with the explanation. I got the answer as B. The question is are all the integers in the set negative? S2 suggests the smallest and the largest are positive. So all the numbers are not negative. So S2 is sufficient. What am I missing?

Bunuel · Answer

Please re-read the solution:
(2) The product of the smallest and largest integers in the set is a prime number.  Since only positive numbers can be primes, then the smallest and largest integers in the set must be of the same sign.  Thus the set consists of only negative or only positive integers .  Not sufficient.

GouthamNandu · Answer

Please elaborate how did you choose option C ? From option B you have concluded that the possibility is only {negative,positive,positive} as prime numbers can only be positive and from option C you have eliminated the same case and choose all numbers to be negative.

Kindly help where I am missing !

Kindly help where I am missing !

Bunuel · Answer

From (1): the set could be either {negative, negative, negative} or {negative, positive, positive}. 
From (2): the set consists of only negative or only positive integers.

(1)+(2) Since the second statement rules out {negative, positive, positive} case which we had from (1), then we have that the set must have only negative integers.

rperaalv · Answer

I think this is a  poor-quality  question and the explanation isn't clear enough, please elaborate. (2) The product of the smallest and largest integers in the set is a prime number.

I don´t see how can this actually happen. The set consist of more than two integers, and the only prime number that can be the product of two integers is 2. Therefore, it could be:
- 1 * - 2 = 2 or (-1 being the largest integer and -2 being the smallest) 
 1 * 2 = 2 (1 being the smallest and 2 being the largest)

However, it says that the set consists of more than two integers.... Please advise

Bunuel · Answer

This could happen VERY easily. Foe example:

{-3, -2, -1} --> The product of the smallest and largest integers in the set = (-3)*(-1) = 3 = prime number.
{-7, -6, -4, -1} --> The product of the smallest and largest integers in the set = (-7)*(-1) = 7 = prime number.
{1, 2, 3, 4, 5, 17} --> The product of the smallest and largest integers in the set = 1*17 = 17 = prime number.
...

Hope now is clear/.

dileeprk · Answer

As the questions says that the product of any three numbers is negative.

Lets say if one number is positive,
+1, -1, -3, -9

As it says any three integers, we can take +1, -1, -3
prod = +3

The only way the product of "ANY" three integers are negative is only if all the numbers in the set are -ve.

The answer should be A.

Bunuel · Answer

The answer should be and is C, not A.

The product of any three integers in the set is negative means that no matter which three integers you pick from the set their product will turn out to be negative. Or in other words ALL sets of three integers you could pick will give negative product. The solution shows a set which gives a negative product while not having all negative terms:  {negative, positive, positive} . The solution also mentions the following:  If the set consists of more than 3 terms, then the set can only have negative numbers .

vaibhav1221 · Answer

Hi  Bunuel !

This question was in my Quant CAT today. I chose A. At the back of my mind, I was confused between A and C. But could not find a good an example. In the question it says that set S has more than 2 integers and according to statement 1, product of any 3 integers is negative. The explanation gives an example where S has 3 integers. Consider the following set
{2,3,-5,-2,-4}
In this set, the product of any 3 integers will be negative if we take odd number of negative integers i.e 1 and 3. However, in a case where we choose {2,-5,-4} or similar 3 integers from set S, where there are even number of negative integers, the product is positive.
Please can you help me identify my mistake.

Thanks,
V

Bunuel · Answer

{2,3,-5,-2,-4} does not satisfy the condition that the product of  ANY  three integers in the set is negative: 2*(-5)*(-2) = 20 = positive.

As explained in the solution, this could be true if:
a. the set consists of only 3 terms, then the set could be either {negative, negative, negative} or {negative, positive, positive}. 
b. the set consists of more than 3 terms, then the set can only have negative numbers.

Pari28 · Answer

I think this is a  high-quality  question and I agree with explanation.

rip2020 · Answer

Hi Bunuel

For statement 1, why are you assuming that the set can have only 3 numbers, what if the set has many more numbers. Then how will you combine the statements?

Please advise.

Thanks!

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