Bunuel wrote:

Is x an even integer?

(1) x^2 is divisible by 4.

(2) x^4 is divisible by 16.

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:The problem is asking whether x is even – that is, whether x is divisible by 2.

Statement (1): NOT SUFFICIENT. It’s easy to look for “confirming” evidence by assuming the question. That’s a bad strategy. Rather, take the statement as true, and see whether you answer the question the same way every time. In particular, with a Yes/No question, be a skeptic. Try to find both a Yes case and a No case.

The statement says that x^2 is divisible by 4. That is, x^2 is a multiple of 4. List a few multiples of 4 and then answer the question using those cases.

First case: x^2 = 4 itself. So then x = 2 or –2. In both scenarios, x is indeed even, so the answer is Yes.

Second case: x^2 = 8. So then x = the positive or negative square root of 8 (√8 or –√8). Since 8 is not a perfect square, x is not an integer (the positive root is between 2 and 3). Only integers can be even, so the answer to the question is No.

Since we have a Yes case and a No case, we can stop. This statement is insufficient.

Statement (2): NOT SUFFICIENT. As in the previous analysis, you should take the statement as true, then apply different cases to the question and see whether you get the same answer each time.

The statement says that x^4 is divisible by 16. That is, x^4 is a multiple of 16. List a few multiples of 16 and then answer the question using those cases.

First case: x^4 = 16 itself. So then x = 2. This is an even integer, so the answer is Yes.

Second case: x^4 = 32. So then x = the fourth root of 32. Since 32 is not a perfect fourth power, x is not an integer, let alone an even integer. So the answer is No.

Statements (1) and (2) together: NOT SUFFICIENT. There are still Yes and No cases. In fact, the first statement implies the second statement.

If x^2 is divisible by 4, then we can write x^2 = 4n, where n is an integer.

Squaring both sides gives us x^4 = 16n^2, and since n^2 is automatically an integer if n is an integer, the second statement is also satisfied.

So the second statement adds nothing to the first statement, which is more restrictive. Every case that satisfies the first statement satisfies the second statement as well. Together, the statements are NOT sufficient.

The correct answer is E.
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