Which of the following statements about prime numbers is not true?

Introduction / Context:
This question tests your understanding of basic facts and theorems about prime numbers. Several statements are given, and you must identify which one is not always true. The key is to recall known properties of primes and to look for a counterexample to any suspicious generalization.

Given Data / Assumptions:
Statement (a): The difference of two primes greater than 2 is divisible by 2.Statement (b): If a prime p divides m × n, then p divides m or p divides n.Statement (c): Every number of the form 6n − 1 (n ∈ N) is a prime.Statement (d): There is only one set of three primes with gaps of 2 between consecutive primes.

Concept / Approach:
We analyze each statement. Statements (a) and (b) are standard facts: (a) follows because all primes greater than 2 are odd, and (b) is a fundamental theorem about primes dividing products. Statement (d) refers to prime triplets. Statement (c) is a strong claim that all numbers of a certain form are prime, which is suspicious because most such simple forms include composite numbers as well.

Step-by-Step Solution:
Step 1: Check statement (a). Any prime greater than 2 is odd. The difference of two odd numbers is even, and every even number is divisible by 2. So statement (a) is always true.Step 2: Check statement (b). This is a well known property: if p is prime and p divides m × n, then p must divide at least one of m or n. This is true for all integers m and n.Step 3: Check statement (d). The only triple of primes of the form p, p + 2 and p + 4 is 3, 5 and 7. For larger primes, at least one of the three consecutive odd numbers is divisible by 3, so they cannot all be prime. Thus there is indeed only one such prime triplet, and statement (d) is true.Step 4: Check statement (c). It claims that all numbers of the form 6n − 1 are prime. Take n = 6; 6n − 1 = 36 − 1 = 35, which equals 5 × 7 and is composite. Therefore statement (c) is not always true.

Verification / Alternative check:
Other counterexamples for statement (c) also exist, such as n = 10 giving 6 × 10 − 1 = 59 (prime) but n = 12 giving 71 (prime) and n = 15 giving 89 (prime), while n = 11 gives 65 = 5 × 13 (composite). The form 6n − 1 includes both primes and composites, so the claim “it is a prime number” for all n is false.

Why Other Options Are Wrong:
Options (a), (b) and (d) describe correct facts about primes and prime gaps. They cannot be the “not true” statement. Only option (c) makes an over-generalized claim that fails when tested with specific values of n.

Common Pitfalls:
Some learners forget that many primes greater than 3 can be written as 6n ± 1 and incorrectly conclude that every number of this form is prime. This is not correct; the form includes many composite numbers. Always look for counterexamples when a statement claims that “every number of this form is prime.”

Final Answer:
The statement that is not always true is “If a number is of the form 6n − 1, then it is a prime number.”