Official Guide Explanation:
Data Sufficiency #31




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Solution and Metadata

Question: 31
Page: 155
Difficulty: 7 (Very Difficult)
Category 1: Arithmetic > Powers and Roots of Numbers > Roots
Category 2: Arithmetic > Properties of Integers > Other

Explanation: As usual, think of other ways to phrase the question: in this case, you want to know whether x is a perfect square.

Statement (1) is sufficient. If rt[4x] is an integer, then its equivalent, 2 rt[x], is an integer. Thus, rt[x] must be one of two things: an integer, or one - half an integer, such as (3/2). However, if x is a number such as (3/2), x is not an integer: x in this case would be ((3/2))2 = (9/4). Since x must be an integer, we can rule out this second group, and reason that x must be an integer.

Statement (2) is not sufficient. If rt[3x] is not an integer, then its equivalent, rt[3] rt[x] must not be an integer. If x = 4, then rt[3x] is not an integer, and rt[x] is an integer, 2. However, if x = 2, then rt[3x] is not an integer, and rt[x] is not an integer. Since rt[x] could be either an integer or a non - integer, the statement isn't sufficient. Choice (A) is correct.

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