If $p$ is a prime number greater than 3, then $(p^2 - 1)$ is always divisible by

## Concept & Logic Any prime number greater than 3 can be expressed in the form $6k \pm 1$. Alternatively, you can analyze the algebraic expansion of the difference of squares: $a^2 - b^2 = (a-b)(a+b)$. ## Step-by-Step Solution * Expand the given expression: $$ p^2 - 1 = (p - 1)(p + 1) $$ * Since $p$ is a prime number greater than 3, $p$ must be an odd number. * Therefore, both $(p - 1)$ and $(p + 1)$ are consecutive even numbers. * In any two consecutive even numbers, one is a multiple of 2 and the other is a multiple of 4. Their product is therefore divisible by $2 \times 4 = 8$. * Additionally, consider the three consecutive integers: $(p - 1)$, $p$, and $(p + 1)$. Exactly one of them must be a multiple of 3. * Since $p$ is a prime $> 3$, $p$ cannot be the multiple of 3. Thus, either $(p - 1)$ or $(p + 1)$ must be divisible by 3. * Combining these factors, the product $(p - 1)(p + 1)$ is perfectly divisible by both 8 and 3, meaning it is divisible by $8 \times 3 = 24$. ## Exam Strategy & Shortcut Use value substitution for instant results. Take the first prime number greater than 3, which is $p = 5$. Calculate $5^2 - 1 = 25 - 1 = 24$. Check the next prime, $p = 7$. Calculate $7^2 - 1 = 49 - 1 = 48$. Both results are divisible by 24, firmly pointing to option (c). ## Common Pitfall Students often pick $p = 2$ or $p = 3$ without reading the constraint "$> 3$". If $p = 3$, $3^2 - 1 = 8$, which is not divisible by 24, leading test-takers to incorrectly choose "None of these". Always honor the constraints in the prompt. ## Final Answer **Therefore, the correct answer is 24.**