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(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.

"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. 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.

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

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

Please re-read the solution for (1): (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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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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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.

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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