Question

For integers w, x, y, and z, is wxyz = -1?
(1) wx/yz = -1
(2) w = -1/x and y = 1/z

• 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.

Hint:

When variables are stated to be integers, equations like \(ax=1\) or \(ax=-1\) severely restrict the possible values for `a` and `x`. This is a powerful constraint to look for in Data Sufficiency problems.

Answer 1

The Correct Option is B

Step 1: Understanding the Question
This is a Yes/No question. We are given that w, x, y, and z are integers. For their product `wxyz` to equal -1, an odd number of them must be -1 and the rest must be 1. In other words, each variable must be either 1 or -1.

Step 2: Analysis of Statement (1)
Statement (1) gives the equation \( wx/yz = -1 \), which implies \( wx = -yz \). Let's test some integer values that satisfy this condition. \[\begin{array}{rl} \bullet & \text{Case 1: Let w=1, x=1, y=1, z=-1.} \\ \bullet & \text{Case 2: Let w=2, x=1, y=-2, z=1.} \\ \end{array}\] Since we can get both a "Yes" and a "No" answer, Statement (1) ALONE is not sufficient.

Step 3: Analysis of Statement (2)
Statement (2) gives two equations: \(w = -1/x\) and \(y = 1/z\). From \(w = -1/x\), we can write \(wx = -1\). Since w and x must be integers, the only possibilities are: \[\begin{array}{rl} \bullet & \text{w = 1 and x = -1} \\ \bullet & \text{w = -1 and x = 1} \\ \end{array}\] In both cases, the product \(wx\) is -1.
From \(y = 1/z\), we can write \(yz = 1\). Since y and z must be integers, the only possibilities are: \[\begin{array}{rl} \bullet & \text{y = 1 and z = 1} \\ \bullet & \text{y = -1 and z = -1} \\ \end{array}\] In both cases, the product \(yz\) is 1.
Now we can find the value of the product wxyz: \[ wxyz = (wx)(yz) = (-1)(1) = -1 \] This gives a definite value of -1. The answer to the question is always "Yes".
Therefore, Statement (2) ALONE is sufficient.
Step 4: Final Answer
Since Statement (2) alone is sufficient, but Statement (1) alone is not, the correct answer is (B).