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 the largest integers in the set is a prime number

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Correct Answer:
Solution with Explanation:
Approach Solution (1):

S1: 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}. If the set consists of more than 3 terms, then the set can only have negative numbers
Not sufficient

S2: The product of the smallest and the largest integers in the set is a prime number. Since only positive numbers can be primes, then the smallest and the largest integers in the set must be 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

Correct Option: C

Approach Solution (2):

Statement 1: The product of any three integers in the list is negative
There are only 2 scenarios in which the product of 3 integers is negative.
scenario #1: all 3 integers are negative
scenario #2: 2 integers are positive, and 1 integer is negative

So, here are two possible cases that satisfy statement 1:
Case a: set S = {-3, -2, -1}, in which case all of the numbers in set S are negative
Case b: set S = {-1, 1, 3}, in which case not all of the numbers in set S are negative
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The product of the smallest and largest integers in the list is a prime number.
Here are two possible cases that satisfy statement 2:
Case a: set S = {-3, -2, -1}, in which case all of the numbers in set S are negative
Case b: set S = {1, 2, 3}, in which case not all of the numbers in set S are negative
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Earlier, we learned that, if the product of 3 integers is negative, then there are 2 possible scenarios:
- scenario #1: all 3 integers are negative
- scenario #2: 2 integers are positive, and 1 integer is negative

Statement 2 tells us that the product of the smallest and largest integers in the list is a prime number. In other word, the product of the smallest and largest integers is POSITIVE.
This allows us to eliminate scenario #2, because under this scenario, the smallest integer in set S would be negative and the largest would be positive, so the product would be NEGATIVE (and prime numbers, by definition, are positive)

This leaves us with scenario #1.
From here, we can conclude that, if the product of any three integers is ALWAYS negative, then ALL of the integers in the set must be negative.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Correct Option: C

Approach Solution (3):

From S1, we have at most two positive integers (if we had three +'s, we'd have + * + * + = +). If we have 0 positives, we have a set of three negative integers. If we have two positives, we have one negative. Since we already have two possible outcomes (all negative or only one negative), INSUFFICIENT.

From S2, we either have (1) * (prime) = prime or (-1) * (-prime) = prime. Since our range (smallest to largest) goes from either pos -> pos or neg -> neg, either ALL our #'s are positive or ALL our #'s are negative.

INSUFFICIENT.

Together, we know from S1 that we have at most two positive integers, but we know from the stem that we have more than two integers. Hence we must have at least one negative, and from S2 we know that ALL our integers are negative.

Correct Option: C

“Set S consists of more than two integers. Are all the integers in set S negative?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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