Question

A number is called even-odd if it is halfway between an even integer and an odd integer. If $x$ is an even-odd number, which of the following must be true?
I. $2x$ is an integer.
II. $2x$ is even-odd.
III. $x$ is halfway between two even integers.

• A. I only
• B. II only
• C. I and II only
• D. II and III only
• E. I, II and III

Answer 1

The correct option is A I only

Let us say the two numbers are $2m+1$ (odd) and $2n$ (even), where $m$ and $n$ are integers irrespective of being odd or even.

Halfway between them is $\frac{(2m+1+2n)}{2}=m+n+\frac{1}{2}$

So the even-odd number $X$ must be of the form $N+\frac{1}{2}$, where $N$ can be any integer.

Therefore

$2$ $X$ is an integer (odd)

$2$ $X$ cannot be an even-odd number since $2$ $X$ is an integer and not of the form $N+\frac{1}{2}$

$X$ cannot be halfway between two even integers, since half way between two even integers is an integer too and $X$ is of the form $N+\frac{1}{2}$

Therefore only the first one is correct which means the answer is A.