Question

$x$, $y$, and $z$ are three positive odd integers. Is $x + z$ divisible by 4?
I. $y - x = 2$
II. $z - y = 2$

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

Hint:

For parity (odd/even) and divisibility problems, combine statements to see full number relationships before concluding.

Answer 1

The Correct Option is C

Odd integers differ by an even number. From Statement I: $y - x = 2$ implies $y = x + 2$. This gives no direct information about $z$, so we cannot decide divisibility of $x+z$ by 4.
From Statement II: $z - y = 2$ implies $z = y + 2$, but without knowing $x$, we again cannot conclude.
Combining both: From I, $y = x + 2$, and from II, $z = y + 2 = x + 4$. Thus, $x + z = x + (x + 4) = 2x + 4$. Since $x$ is odd, $2x$ is even, and $2x + 4$ is divisible by 2 but not by 4 (because $2x$ is not a multiple of 4 when $x$ is odd). Therefore, the answer is definitive: NO.
Hence, both statements are required.