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.