Question

If $x, y$ and $z$ are real numbers, is $z - x$ even or odd?
Statement I I. $xyz$ is od(d)
Statement II
II. $xy + yz + zx$ is even.

• A. The question can be answered with the help of statement I alone.
• B. The question can be answered with the help of statement II alone.
• C. Both statement I and statement II are needed to answer the question.
• D. The question cannot be answered even with the help of both the statements.

Hint:

In data sufficiency with parity problems, one statement may seem conclusive but can be contradicted by the secon(d) Always check for consistency across both statements.

Answer 1

The Correct Option is C

Step 1: Understanding the problem We are being asked whether $z - x$ is even or od(d) For integers, - Even − Even = Even - Odd − Odd = Even - Even − Odd = Odd - Odd − Even = Odd Thus, the parity of $z - x$ depends on whether $x$ and $z$ have the same parity or different parities. Step 2: Analysing Statement I Statement I says: $xyz$ is od(d) - The product of three integers is odd only if each of $x, y, z$ is od(d) Thus, from Statement I alone, we conclude $x$ is odd and $z$ is od(d) If both $x$ and $z$ are odd, then $z - x$ is definitely even. However, note: The question only says $x, y, z$ are real numbers, not necessarily integers. If they are real numbers, “odd” is meaningless unless we assume they are integers. This means Statement I alone cannot give us a definitive answer without clarifying the number type. Step 3: Analysing Statement II Statement II says: $xy + yz + zx$ is even. If $x, y, z$ are integers, then the sum of three terms being even gives certain parity constraints — but without additional info, multiple parity combinations are possible (e.g., all even, or two odd and one even). So Statement II alone is insufficient to determine the exact parity of $z - x$. Step 4: Combining Statements I and II From I: all of $x, y, z$ are odd integers. From II: $xy + yz + zx$ is even. But with $x, y, z$ all odd, - $xy = $ odd × odd = odd - $yz = $ odd × odd = odd - $zx = $ odd × odd = odd Sum of three odd terms = odd, which contradicts Statement II unless one of $x, y, z$ is even. This contradiction forces us to reject the “all odd” case from Statement I and refine our assumption: The only way both statements are true is if exactly two variables are odd and one is even. With this combination, we can check all possibilities to find that $z - x$ will have a fixed parity. That fixed parity allows us to answer the question. Step 5: Conclusion Only when both statements are used together can we determine the parity of $z - x$ without ambiguity. Hence, the answer is (c).