If $n$ and $p$ are both odd numbers, which of the following is an even number?

### Concept & Logic Parity rules dictate how even and odd numbers interact during basic arithmetic operations. The sum of two odd numbers is always an even number. ### Step-by-Step Solution **Given:** * $n$ is an odd number. * $p$ is an odd number. **Calculation / Deduction:** Let's evaluate the options using the fundamental rules of parity: * **Option (a) $n + p$**: The sum of two odd numbers is always even (Odd + Odd = Even). * **Option (b) $n + p + 1$**: Since $n + p$ is even, adding 1 to an even number results in an odd number (Even + 1 = Odd). * **Option (c) $np + 2$**: The product of two odd numbers ($np$) is odd. Adding an even number (2) to an odd number results in an odd number (Odd + Even = Odd). * **Option (d) $np$**: The product of two odd numbers is always odd (Odd $\times$ Odd = Odd). ### Exam Strategy & Shortcut Instead of thinking abstractly, instantly substitute the smallest positive odd numbers to test the options. Let $n = 1$ and $p = 3$. (a) $1 + 3 = 4$ (Even) (b) $1 + 3 + 1 = 5$ (Odd) (c) $(1 \times 3) + 2 = 5$ (Odd) (d) $1 \times 3 = 3$ (Odd) Option (a) is the only even result. ### Common Pitfall A common pitfall is wasting time trying to prove the algebra (e.g., setting $n = 2k+1$ and $p = 2m+1$) when a simple substitution method provides an immediate, foolproof verification of parity rules under time constraints. ### Final Answer Therefore, the correct answer is **$n + p$**.