Official Guide Explanation:
Data Sufficiency #70

Background

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Solution and Metadata

Question: 70
Page: 158
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Real Numbers >
Category 2: Arithmetic > Properties of Integers > Factors and Multiples

Explanation: This is an unusual, tricky question. It takes some experimentation to figure out what the rules mean. Let's say 5 is in K. Then -5 is in K as well. Based on (ii), if 5 and -5 are in K, then -25 is in K. Add another number, and the possibilities multiply, quite literally.

Statement (1) is insufficient. If 2 is in K, -2 is in K, as is -4, as well as 8 and -8. Note the pattern: These are all positive and negative powers of 2. 12 is not a power 2, so we don't know that 12 is in K. However, there could be other numbers besides the powers of 2, so we can't answer the question.

Statement (2) is insufficient for the same reason. Here we can only be confident about the presence of powers of 3. We know 3, -3, 9, 27, -27, etc. are present.

Taken together, the statements are sufficient. (1) established that -4 is in K, while (2) told us that -3 is in K. Since if two numbers are in K, their product is in K as well (rule (ii)), 12 is in K. Choice (C) is correct.

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