Manhattan GMAT Challenge Problem of the Week – 6 May 10
Welcome back to this weeks Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!
If x and y are integers and x < y, what is the value of x + y?
(1) [pmath]x^y[/pmath] = 4
(2) |x| = |y|
(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.
The question cannot be easily rephrased to incorporate the particular information given. However, of course we should take note that both variables are integers and that x is less than y. We are looking for the value of x + y.
Statement (1): SUFFICIENT. First, we should list out all the possible scenarios in which integers x and y fit the equation [pmath]x^y[/pmath] = 4.
There are three possibilities, as we can find by trial and error: [pmath]2^2[/pmath] = 4, [pmath](-2)^2[/pmath] = 4, and [pmath]4^1[/pmath] = 4. However, of these possibilities, there is only one for which x is less than y, namely [pmath](-2)^2 = 4[/pmath]. Thus, we can find the value of x + y, which is -2 + 2 = 0.
Statement (2): SUFFICIENT. Knowing that |x| = |y| does not tell us the values of the integers. However, since they have the same absolute value, but x is less than y, it must be the case that y is a positive integer and x is the negative of that integer. For instance, if y is 5, then x is -5. The sum of x and y must therefore be 0, no matter what.
The correct answer is (D).
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