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

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Set S consists of more than two integers. Are all the numbers in set S [#permalink]

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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
(2) The product of the smallest and largest integers in the list is a prime number.
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Re: Set S consists of more than two integers. Are all the numbers in set S [#permalink]

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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.
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Re: Set S consists of more than two integers. Are all the numbers in set S [#permalink]

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ZArslan wrote:
24. 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
(2) The product of the smallest and largest integers in the list is a prime number.


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.
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Re: Set S consists of more than two integers. Are all the numbers in set S [#permalink]

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New post 16 Jan 2018, 22:51
ZArslan wrote:
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
(2) The product of the smallest and largest integers in the list is a prime number.


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 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.

Answer: C.
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Re: Set S consists of more than two integers. Are all the numbers in set S   [#permalink] 16 Jan 2018, 22:51
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