Manhattan GMAT Challenge Problem of the Week - 12 April 2011

by Manhattan Prep, Apr 12, 2011

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Question

If xy 0, is x an integer?

1) [pmath]x^y[/pmath] is an integer.

2) y is a prime number with y unique positive factors.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer

The question is asking whether x is an integer. Glancing at the statements, we can see that statement (2) only involves y, so it cannot be sufficient. (All we are told in the stem is that xy is not equal to 0, another way of saying that neither x nor y is equal to 0. Thats not a lot of information.) So we can rule out B and D without even analyzing (2) closely.

Statement (1) tells us that [pmath]x^y[/pmath] is an integer. This does not guarantee that x is an integer. Of course, it could be. If y = 1, for instance, then [pmath]x^y[/pmath] = x , so if [pmath]x^y[/pmath] is an integer, then x is as well. However, if y = -2, then x could be , which is not an integer. raised to the -2 power is 4. So we can rule out A.

Now, put the two pieces of information together. We are told in (2) that y is a prime number with y unique factors. But every prime number has just 2 unique positive factors (1 and itself), so y must be 2. Combining this fact with the other statement, we know that [pmath]x^2[/pmath] is an integer. Does this tell us whether x is an integer? Of course, x could be an integer ([pmath]2^2[/pmath] = 4), but the question is really whether x has to be an integer. The answer is no. After all, [pmath]x^2[/pmath] could equal 3. Then x would be equal to the square root of 3. Knowing that the square of x is an integer does not guarantee that x itself is an integer.

The correct answer is E.

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