Question

If $x,y$ and $z$ are positive integers, is $x-y$ odd?
(1) $x=z^2$
(2) $y=(z-1{)}^2$

• 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.

Answer 1

The correct option is C Both statements together are sufficient, but neither statement alone is sufficient.

1. This reveals the relationship between two of the variables but does not mention the relationship either has with y. Therefore the question cannot be answered; NOT sufficient.
2. If is expanded, the result is . Since , a substitution for y can be made in the expression $(x-y)$. It becomes . However, without further information, it cannot be determined if $(x-y)$ is odd; NOT sufficient.

• When (1) and (2) are taken together, , from (1), can be substituted for x in the expression from (2). It then yields , or simply $(2z-1)$, which is always an odd number regardless of the value of $z$. So $(x-y)$ is odd.
• The correct answer is C; both statements together are sufficient.