Hi jswesth,

This DS question involves 'causality', which is a relatively rare concept in the Quant section (you probably will NOT see it there on Test Day, but you WILL see causality in a couple of CR prompts in the Verbal section).

We're asked if the number 12 is in Set K. This is a YES/NO question.

1) 2 is in K.

Based on the information in this Fact, we can determine a variety of numbers in Set K, including...

2 and -2 (first rule)
(2)(-2) = -4 (second rule)
-4 and 4 (first rule)
(2)(4) = 8 (second rule)
(2)(-4) = -8 (second rule)
Etc.

From these examples, you should notice that ALL of the numbers are some 'power of 2'; 12 is NOT a power of 2 though, so we don't know if 12 is actually in Set K or not.
Fact 1 is INSUFFICIENT

2) 3 is in K.

The same logic that we used in Fact 1 can be applied here (just with the number 3 instead of the number 2)...

3 and -3 are in Set K (first rule)
(3)(-3) = -9 (second rule)
-9 and 9 (first rule)
(3)(-9) = -27 (second rule)
Etc.

From these examples, you should notice that ALL of the numbers are some 'power of 3'; 12 is NOT a power of 3 though, so we don't know if 12 is actually in Set K or not.
Fact 2 is INSUFFICIENT

Combined, we know LOTS of numbers that are in Set K. From all of our prior work, we know that 3 and 4 are BOTH in Set K, so (3)(4) = 12 is ALSO in Set K. The answer to the question is ALWAYS YES.
Combined, SUFFICIENT.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

jswesth wrote:K is a set of numbers such that:
(i) If x is in K, then -x is in K, and
(ii) if each x and y is in K, then xy is in K.

Is 12 in K?

(1.) 2 is in K.
(2.) 3 in is K.

Target question: Is 12 in K?

Statement 1: 2 is in K
By rule i, -2 is also in set K
By rule ii, if 2 and -2 are in set K, then -4 is in set K
By rule i, 4 is also in set K
By rule ii, if 2 and -4 are in set K, then -8 is in set K
By rule ii, if 2 and 4 are in set K, then 8 is in set K
etc..
So, all we can say is that the following numbers MUST be in K: ...-16, -8, -4, -2, 2, 4, 8, 16...
Since we don't know what other numbers might be in K, 12 might be in K or 12 might NOT be in K.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 3 is in K
By rule i, -3 is also in set K
By rule ii, if 3 and -3 are in set K, then -9 is in set K
By rule i, 9 is also in set K
By rule ii, if 3 and -9 are in set K, then -27 is in set K
By rule ii, if 3 and 9 are in set K, then 27 is in set K
etc..
So, all we can say is that the following numbers MUST be in K: ...-27, -9, -3, 3, 9, 27,...
Since we don't know what other numbers might be in K, 12 might be in K or 12 might NOT be in K.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that 4 must be in set K
Statement 2 tells us that 3 must be in set K
By rule ii, if 4 and 3 are in set K, then 12 is in set K
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent