Question

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let $p$ be a prime number greater than 5. Then $(p^2 - 1)$ is:

• A. never divisible by 6
• B. always divisible by 6, and may or may not be divisible by 12
• C. always divisible by 12, and may or may not be divisible by 24
• D. always divisible by 24

Hint:

Use factorization and properties of consecutive integers to prove divisibility of expressions involving primes.

Answer 1

The Correct Option is D

Let $p$ be a prime number $>5$ Then $p$ is odd and not divisible by 3 \[ p^2 - 1 = (p - 1)(p + 1) \] Now, among 3 consecutive even numbers: $p - 1$, $p$, $p + 1$ ⇒ $(p - 1)(p + 1)$ is the product of two even numbers: - One divisible by 2 - One divisible by 4 (as two even numbers are 2 apart) Also, among any 3 consecutive integers, one is divisible by 3 Since $p$ is not divisible by 3 (as it’s a prime>5), either $p - 1$ or $p + 1$ is divisible by 3 So: \[ (p - 1)(p + 1) \text{ is divisible by } 2 \cdot 4 \cdot 3 = \boxed{24} \]