Question

If $x$ and $y$ are positive integers such that $x$ is an odd prime and $y$ is composite, then which of the following is true?

• A. $y-x$ cannot be an even integer
• B. $xy$ cannot be an even integer
• C. $\frac{(x+y)}{x}$ cannot be an even integer
• D. $x+y$ is an integer

Answer 1

Answer: (D)

$x+y$ is an integer

Given that, $x$ is prime and $y$ is composite.

$\Rightarrow x$ is an odd number.

Option A: $⟶y-x$ will be even, if $y$ is odd.

$\therefore y-x$ can be an even integer.

$\therefore$ option A is not true.

Option B: $⟶xy$ is an even number, if $y$ is even.

$\therefore$ option B is not true.

Option C: $⟶\frac{x+y}{xy}$

$x+y$ may be even and $x+y$ is even and $xy$ may have a factor of $x+y$ and another even number.

So, option C is not true.

Option D: $⟶x$ is an integer and $y$ is an integer.

So, their sum will be an integer.

$\therefore$ option D is correct.