psm12se 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.

Target question: Are all the numbers in set S negative?

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

Answer = C

Cheers,
Brent

psm12se wrote: I am confused about the prime numbers. Is it possible to have a -ve prime number in which case answer would be E

Prime numbers, by definition, are positive.

Here's how the OG defines prime numbers: A prime number is a positive integer that has exactly two different positive divisors, 1 and itself.

Cheers,
Brent