Question

Let $x$ and $y$ be positive integers such that $x$ is prime and $y$ is composite. Then,

• A. $1 - y - x$ cannot be an even integer
• B. $xy$ cannot be an even integer
• C. $\frac{x + y}{x}$ cannot be an even integer
• D. None of these

Hint:

When dealing with composite and prime numbers, check for divisibility and simplify expressions to test for even or odd properties.

Answer 1

The Correct Option is C

Let $x$ be a prime number, and $y$ be a composite number. To check each option:
1. The expression $1 - y - x$ involves subtracting two integers, and its parity depends on $x$ and $y$. It is not guaranteed to always produce an odd or even number.
2. Since $x$ is prime, and $y$ is composite, we can check the parity of $xy$. If $x$ is odd and $y$ is even, $xy$ can still be even.
3. The expression $\frac{x + y}{x}$ simplifies to $1 + \frac{y}{x}$. Since $y$ is composite, $y$ might not be divisible by $x$, making the whole expression not an even integer.
Thus, the Correct Answer is option (3).