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# M28-53

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Math Expert
Joined: 02 Sep 2009
Posts: 55632

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16 Sep 2014, 01:44
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95% (hard)

Question Stats:

40% (01:10) correct 60% (01:12) wrong based on 202 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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16 Sep 2014, 01:44
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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07 May 2015, 09:55
"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."

why does the set consist of only negative or only positive integers? It could be that only the smallest and the largest have the same sign and not the rest.
It doesnt say the other numbers are also primes.
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07 May 2015, 10:06
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1
shly wrote:
"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."

why does the set consist of only negative or only positive integers? It could be that only the smallest and the largest have the same sign and not the rest.
It doesnt say the other numbers are also primes.

(2) says: 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.

So either: smallest * greatest = negative * negative and in this case as both the smallest and the greatest are negative then ALL integers in the list are negative OR smallest * greatest = positive * positive and in this case as both the smallest and the greatest are positive then ALL integers in the list are positive.

Hope it's clear.
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08 Jul 2016, 11:20
It says the product of any three integers in set is Negative. So, even if there are two positive numbers this is not possible. Because what if we choose of the positives and rest two negatives. And the statement says that product is always negative. So I think A is sufficient because all should be negatives
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08 Jul 2016, 12:47
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Navinder wrote:
It says the product of any three integers in set is Negative. So, even if there are two positive numbers this is not possible. Because what if we choose of the positives and rest two negatives. And the statement says that product is always negative. So I think A is sufficient because all should be negatives

(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.
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18 Jul 2016, 10:40
I don't agree with the explanation. If the set consists of only 3 integers, then A+B could give two possibilities:

(negative,negative,negative)
or
(positive, negative,positive)

Both the cases satisfy the two conditions. So the answer should be E
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18 Jul 2016, 10:42
sanjay452 wrote:
I don't agree with the explanation. If the set consists of only 3 integers, then A+B could give two possibilities:

(negative,negative,negative)
or
(positive, negative,positive)

Both the cases satisfy the two conditions. So the answer should be E

Hope it helps.
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06 Jun 2017, 19:34
Bunuel wrote:

So either: smallest * greatest = negative * negative and in this case as both the smallest and the greatest are negative then ALL integers in the list are negative OR smallest * greatest = positive * positive and in this case as both the smallest and the greatest are positive then ALL integers in the list are positive.

Hope it's clear.

Bunuel , if both are negative then middle number is negative as well (so all are negative)
If both are positive then middle number is positive and all are positive.

Doesn't that mean it satisfies both conditions and hence its not sufficient. Or am i completely wrong about DS criteria ?
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25 Nov 2017, 22:29
So either: smallest * greatest = negative * negative and in this case as both the smallest and the greatest are negative then ALL integers in the list are negative OR smallest * greatest = positive * positive and in this case as both the smallest and the greatest are positive then ALL integers in the list are positive.

Hope it's clear.

Hi Bunuel,
Is there any compulsion that a set should be arranged in either ascending order or descending order(even if not mentioned in Q specifically).
Like 1st and last terms negative means in between all terms should be less the highest and more than the lowest.
If yes I agree with your above explanation. If not, why cant we consider the case {+ve -ve +ve}.
Can you pls clarify my doubt.
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26 Nov 2017, 02:05
venkys1 wrote:
So either: smallest * greatest = negative * negative and in this case as both the smallest and the greatest are negative then ALL integers in the list are negative OR smallest * greatest = positive * positive and in this case as both the smallest and the greatest are positive then ALL integers in the list are positive.

Hope it's clear.

Hi Bunuel,
Is there any compulsion that a set should be arranged in either ascending order or descending order(even if not mentioned in Q specifically).
Like 1st and last terms negative means in between all terms should be less the highest and more than the lowest.
If yes I agree with your above explanation. If not, why cant we consider the case {+ve -ve +ve}.
Can you pls clarify my doubt.

My friend you did not read carefully enough.

First of all, a set is a collection of elements without any order, while for example a sequence is ordered.

Next, (2) says: The product of the smallest and largest integers in the set is a prime number. Does it say first and last? NO.
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07 Apr 2018, 06:33
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Bunuel

Hi,

I am still getting E as the answer.

Lets say the set has three integers: x, y and z.

(1) xyz = negative
This means {positive, positive, negative} or {negative, negative, negative}. Therefore, INSUFF.

(2) Assuming x is the smallest integer and z the largest integer: xz= prime since prime cannot be negative xz= positive
This means {positive, positive} or {negative, negative}. Therefore, INSUFF.

(1) and (2) Together, xyz can be {positive, negative, positive} or {negative, negative, negative}. Therefore, C. Why does (2) mean ALL integers in the set are either positive or negative? Isn't it only saying the smallest and largest are either both positive or both negative?

Thanks!

Nevermind, just got it. Since (2) indicates that if the smallest and largest are both positive then all numbers in the middle must be positive and if the smallest and largest are both negative then all numbers in the middle must be negative. I just hope I do not overlook such details during the actual test.
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19 Dec 2018, 07:51
1
Hi Bunuel, I tried understanding the solution, but I still having some difficulty on why E is incorrect.

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.
I could choose [-1, -2, -3] or [1, -2, 3] hence this is insufficient as both lead to a product which is negative

(2) The product of the smallest and largest integers in the set is a prime number.
I could choose [-1, -2, -3] or [1,-2, 3] hence this is insufficient as both lead to a product that is a prime number.

Together, either all could be negative, which meets the condition, or the middle number [-2] could be negative, thus leading to E.

Can you kindly guide me where I am going wrong with this approach please?
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19 Dec 2018, 19:36
1
nausherwan wrote:
Hi, I tried understanding the solution, but I still having some difficulty on why E is incorrect.

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.
I could choose [-1, -2, -3] or [1, -2, 3] hence this is insufficient as both lead to a product which is negative

(2) The product of the smallest and largest integers in the set is a prime number.
I could choose [-1, -2, -3] or [1,-2, 3] hence this is insufficient as both lead to a product that is a prime number.

Together, either all could be negative, which meets the condition, or the middle number [-2] could be negative, thus leading to E.

Can you kindly guide me where I am going wrong with this approach please?

Hey. I'm no expert like Bunnel but I can try explaining it to you. Your analysis for statement 1 is correct and definitely insufficient. However, you made a small mistake in one of your examples you provided for statement 2.

Statement 2 says "The product of the smallest and largest integers in the set is a prime number".
Your first example [-1, -2, -3] is valid because the product of -3 and -1 is 3, which is a prime number. However, your next example of [1, -2, 3] is not valid. In this example you provided, -2 is the smallest integer and 3 would be the largest integer and when multiplied together, their product is -6, which is not a prime number. You can only use examples that agree with the condition of the statement. So you cannot use that example to prove whether statement 2 is sufficient or not.
You can pick numbers such as [1, 2, 3] where 1 is the smallest and 3 is the largest, and their product gives you a value of 3, which is a prime number. Therefore, using your first set of numbers [-1, -2, -3] and this new set of numbers [1, 2, 3], you can confidently say statement 2 is insufficient. And with this analysis of statement (2), you can conclude that the integers have to be either all positive or all negative.

Combining the statements together makes it sufficient to answer the question. Knowing the information from statement (2) that the integers in the set are either all positive or all negative and from statement (1) that the product of any 3 of the integers are negative (meaning that the set must have one or more negative integers for the product to be negative, you can conclude that all the integers in the set are negative.
Re: M28-53   [#permalink] 19 Dec 2018, 19:36
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# M28-53

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