S is a set containing 9 different positive odd primes

Hi, there! I'm happy to help with this.

What is the source of this question? This is a very challenging question. It seems to me considerably harder than what the GMAT would ask. It is certainly not something you should worry about figuring out completely in under two minutes!

Here is my explanation. The question states: "S is a set containing 9 different positive odd primes. T is a set containing 8 different numbers, all of which are members of S. Which of the following statements CANNOT be true?"

(A) The median of S is prime.
This must be true. If there are an odd number of members of a set, then the median is a member of the set: it's the middle number, when all the numbers are ranked from smallest to biggest. Every number in S is a positive odd prime, so the median is one of them, and is prime.

(B) The median of T is prime.
This may or may not be true. If a set has an even number of members, the median is average of the two numbers in the middle, when ranked from smallest to biggest. The average of two odd numbers could be even (average of 71 and 73 is 72), and hence not prime, or it could be odd (the average of 71 and 79 is 75). For particularly well chosen odd numbers, the average can be not only odd but also prime -- for example, the average of 89 and 113 is 101, another prime number. If the two middle numbers of T were 89 and 113, the median would be 101, a prime number.

(C) The median of S is equal to the median of T.
Under most configurations for S and T, this wouldn't happen. If you weren't trying to make it happen, it would be unlikely to happen by chance. BUT, if the number dropped going from from S to T was the median of S (say, 101), and if the two middle numbers of T happen to have an average of that number that was dropped (for example, if the two numbers were 89 and 113), then the medians would be equal. In other words, the three middle numbers of S would have to be {. . ., 89, 101, 133, . . .}, and when 101 is dropped in going to T, the median of two would be the average of 89 & 113, which happens to be 101. It's an exceptional case, but it could be true.

(D) The sum of the terms in S is prime.
This may or may not be true. The sum of 9 odd number must be an odd number. That odd number could be prime. For example, the sum of the first nine odd prime numbers {3, 5, 11, 13, 17, 19, 23, 29} is 127, which is prime. If you drop 3 and include the next prime, 31, the set {5, 11, 13, 17, 19, 23, 29, 31} has a sum of 155, which is clearly not prime.

(E) The sum of the terms in T is prime.
This must be false. The sum of eight odd numbers must be an even number. Only 2 is prime, and all other even numbers are not. Therefore, the sum of eight odd prime numbers will be an even number bigger than two, and absolutely cannot be prime.

Again, I realize that's not an incredibly fast approach, but this is a difficult question. Here's another practice question about prime numbers.

https://gmat.magoosh.com/questions/850

The question at that link should be followed by a video solution.

I hope my response was helpful to some extent. Please let me know if you have any questions on what I've said.

Mike