Let $p$ and $q$ be distinct prime numbers greater than $3$. Which of the following statements is/are ALWAYS TRUE?
Options:
(A) $p^2 - 1$ is divisible by $24$.
(B) $p^2 - q^2$ is divisible by $24$.
(C) $p + q$ is always divisible by $6$.
(D) The product $p \times q$ is odd.
Answer Key:
A, B, D
Step-by-Step Solution & Explanation
**Method 1: Modular Number Theory Proof**
1. Any prime $p > 3$ is of the form $6k \pm 1$:
$p^2 - 1 = (6k \pm 1)^2 - 1 = 36k^2 \pm 12k = 12k(3k \pm 1)$
If $k$ is even, $12k$ is a multiple of 24. If $k$ is odd, $3k \pm 1$ is even, so $12 \times (\text{even}) = 24m$. Hence $p^2 - 1$ is always divisible by 24. (A is True).
2. $p^2 - q^2 = (p^2 - 1) - (q^2 - 1)$. Since both terms are divisible by 24, their difference is also divisible by 24. (B is True).
3. Counterexample for (C): Let $p = 5$ and $q = 11$. $p + q = 16$, which is not divisible by 6. (C is False).
4. All primes greater than 2 are odd. The product of two odd numbers is always odd. (D is True).
Therefore, **A**, **B**, and **D** are always true.