If $m, n, o, p$ and $q$ are integers, then $m (n + o) (p - q)$ must be even when which of the following is even?

### Concept & Rule The product of multiple integers is even if at least **one** of the integers (or factors) in the multiplication is even. ### Step-by-Step Solution * **Given:** The expression is $m \times (n + o) \times (p - q)$. This represents three distinct factors multiplied together. * **Deduction:** For the entire product to *always* be even regardless of the other variables, one of the direct factors must definitively be even. * Let's evaluate the options: * If $p$ is even (Option b), $q$ could be odd, making $(p - q)$ odd. If the other factors are odd, the product is odd. * If $m + n$ is even (Option c) or $n + p$ is even (Option d), it doesn't guarantee that any of the explicit factors $m$, $(n + o)$, or $(p - q)$ are even. * If $m$ is even (Option a), then an even number is being multiplied by the rest of the expression. $Even \times Integer = Even$. ### Exam Strategy & Shortcut Identify the standalone multipliers in the expression. Here, $m$ is a standalone factor. If a standalone factor is even, the entire product is mathematically forced to be even, making it the most direct and undeniable condition. ### Common Pitfall Overthinking the properties of addition/subtraction inside the parentheses (like trying to figure out if $n+o$ can be forced to be even). Focus on the overarching operation, which is multiplication. ### Final Answer Therefore, the correct answer is **$m$**.