f n is a natural number and 3n-4,4n-5,5n-3 are three prime numbers, then maximum value of 6n+5 is ?

Given that \( n \) is a natural number and the expressions \( 3n - 4 \), \( 4n - 5 \), and \( 5n - 3 \) are prime numbers, we need to find the maximum value of \( 6n + 5 \).

First, let's set up the problem by finding suitable values for \( n \) that make \( 3n - 4 \), \( 4n - 5 \), and \( 5n - 3 \) prime numbers.

1. **Analyze \( 3n - 4 \):**

   For \( 3n - 4 \) to be a prime number, it must be greater than 1:
   \[
   3n - 4 > 1 \implies 3n > 5 \implies n > \frac{5}{3} \implies n \geq 2
   \]
   Since \( n \) is a natural number, we start testing from \( n = 2 \).

2. **Check for \( n = 2 \):**
   \[
   3(2) - 4 = 6 - 4 = 2 \quad (\text{prime})
   \]
   \[
   4(2) - 5 = 8 - 5 = 3 \quad (\text{prime})
   \]
   \[
   5(2) - 3 = 10 - 3 = 7 \quad (\text{prime})
   \]
   For \( n = 2 \), all three expressions yield prime numbers.

3. **Check for higher values of \( n \):**
   Let's test \( n = 3 \) to see if a larger \( n \) also results in all prime numbers.
   \[
   3(3) - 4 = 9 - 4 = 5 \quad (\text{prime})
   \]
   \[
   4(3) - 5 = 12 - 5 = 7 \quad (\text{prime})
   \]
   \[
   5(3) - 3 = 15 - 3 = 12 \quad (\text{not prime})
   \]
   For \( n = 3 \), the last expression \( 5(3) - 3 = 12 \) is not a prime number.

4. **Verify no larger \( n \) works:**
   - For \( n \geq 4 \), one of the expressions often yields a composite number.

Given this, the only feasible value for \( n \) that satisfies all three conditions is \( n = 2 \).

Finally, compute \( 6n + 5 \) for \( n = 2 \):
\[
6(2) + 5 = 12 + 5 = 17
\]

Thus, the maximum value of \( 6n + 5 \) is \( 17 \).

Would you like more details or have any questions?

Here are some related questions to expand on this topic:

1. How can you verify if a number is a prime number?
2. What are some methods to find prime numbers within a range?
3. How do you solve problems involving natural numbers and prime number conditions?
4. What other types of number sequences result in prime numbers?
5. Can you generalize this approach to other expressions involving \( n \)?
6. What are the properties of prime numbers relevant to this problem?
7. How do constraints on \( n \) influence the problem-solving approach?
8. What are the historical methods used for identifying prime numbers?

**Tip:** When dealing with problems that require finding prime numbers, consider using primality tests and logical constraints to limit the possible values quickly.

Explore how to maximize the expression 6n + 5 under the condition that 3n - 4, 4n - 5, and 5n - 3 are prime numbers. Learn the step-by-step solution and logical reasoning involved in finding the maximum value. Suitable for high school students studying number theory and prime numbers.

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