Question

If $x$ and $y$ are integers, is $xy$ even?
(1) $x=y+1$
(2) $\frac{x}{y}$ is an even integer.

• 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 D Each statement alone is sufficient.

• Determine if xy is even. (1) Since x and y are consecutive integers, one of these two numbers is even, and hence their product is even. For example, if x is even, then x = 2m for some integer m, and thus xy = (2m)y = (my)2, which is an integer multiple of 2, so xy is even; SUFFICIENT.
  
• If is even, then for some integer n, and thus x = 2ny. From this it follows that , which is an integer multiple of 2, so xy is even; SUFFICIENT.
• The correct answer is D; each statement alone is sufficient.