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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   95% (hard)

Question Stats: 35% (01:50) correct 65% (01:35) wrong based on 325 sessions

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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.

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Math Expert V
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Official Solution:

(1) The product of any three integers in the set is negative. If the set consists of only 3 terms, then the set could be either {negative, negative, negative} or {negative, positive, positive}. If the set consists of more than 3 terms, then the set can only have negative numbers. 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, 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.

(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. Sufficient.

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Bunuel wrote:
Official Solution:

(1) The product of any three integers in the set is negative. If the set consists of only 3 terms, then the set could be either {negative, negative, negative} or {negative, positive, positive}. If the set consists of more than 3 terms, then the set can only have negative numbers. 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, 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.

(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. Sufficient.

Hello!

I didnt get this one. What if the set has 4 values - 2 positive and 2 negative.

-> [2,3,-1,-5]

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. Thanks!
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Bunuel wrote:
Official Solution:

(1) The product of any three integers in the set is negative. If the set consists of only 3 terms, then the set could be either {negative, negative, negative} or {negative, positive, positive}. If the set consists of more than 3 terms, then the set can only have negative numbers. 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, 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.

(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. Sufficient.

I didn't get it how is it C ?

and the order can be { positive, negative , positive } rite then how is c correct ?
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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.
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aimtoteach wrote:
Bunuel wrote:
Official Solution:

(1) The product of any three integers in the set is negative. If the set consists of only 3 terms, then the set could be either {negative, negative, negative} or {negative, positive, positive}. If the set consists of more than 3 terms, then the set can only have negative numbers. 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, 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.

(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. Sufficient.

Hello!

I didnt get this one. What if the set has 4 values - 2 positive and 2 negative.

-> [2,3,-1,-5]

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. Thanks!

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.
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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
Math Expert V
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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

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.
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Consider the sets {1,-2, 5} and {-1,-2,-5}. These sets satisfy both St.1 and 2, and so you cannot definitively answer the question. Ans E for me. Not satisfied with the explanation
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sidney123 wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Consider the sets {1,-2, 5} and {-1,-2,-5}. These sets satisfy both St.1 and 2, and so you cannot definitively answer the question. Ans E for me. Not satisfied with the explanation

You did not understand the question.

(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, 5} is -2*5 = -10, which is NOT prime!
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I think this the explanation isn't clear enough, please elaborate. in stat. 2 it doesnt have to mean that all the integers in the set are positive. i can be
-ve +ve -ve
or
+ve -ve +ve (just an example)
we cant say that all the int. have to be of the same sign.
Math Expert V
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nishantdoshi wrote:
I think this the explanation isn't clear enough, please elaborate. in stat. 2 it doesnt have to mean that all the integers in the set are positive. i can be
-ve +ve -ve
or
+ve -ve +ve (just an example)
we cant say that all the int. have to be of the same sign.

The smallest integer in your examples above would be negative and the largest one would be positive giving negative product, which cannot be a prime.
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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?
Math Expert V
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njj1984 wrote:
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?

(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.
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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 !
Math Expert V
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GouthamNandu wrote:
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 !

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.
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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
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rperaalv wrote:
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

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/.
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I think this is a poor-quality question and the explanation isn't clear enough, please elaborate.
Math Expert V
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karn99 wrote:
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate.

I'd suggest to read the whole discussion once again and ask specific question.

P.S. The question is 100% correct and up to highest GMAT standards. It's not that easy though for novices.
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