Math Problem Statement

is this correct ? Given the problem, we need to determine the maximum value of nn such that both n+5* is a twin prime and n+8 is not a prime number, and satisfies the inequality n+5n−5>2.

Step 1: Consider the inequality n+5n−5>2

We start by solving the inequality:

n+5n−5>2

This can be rewritten as:

n+5>2(n−5)

Simplifying the right side:

n+5>2n−10

Bringing nn terms on one side:

5+10>2n−n

15>n

Thus we have:

n<15

Step 2: Identify n+5 as a twin prime.

A twin prime p fulfills (p−2) or (p+2) also being prime. Let's list twin primes to find eligible n+5:

(3,5)

(5,7)

(11,13)

(17,19)

Matching these with n+5:

n+5=5 leads to n=0

n+5=7 leads to n=2

n+5=11 leads to n=6

n+5=13 leads to n=8

n+5=17 leads to n=12

n+5=19 leads to n=14

Step 3: Ensure n+8 is not a prime number.

Checking listed values:

For n=0, n+8=8 (not prime).

For n=2, n+8=10 (not prime).

For n=6, n+8=14 (not prime).

For n=8, n+8=16 (not prime).

For n=12, n+8=20 (not prime).

For n=14, n+8=22 (not prime).

All listed n+8values for n* from 0 to 14 are not prime. The largest possible n* while fulfilling all constraints is:

n=14

Thus, the maximum possible value of n is:

14

Solution

The reasoning and calculations appear mostly correct, but let's review the logic step by step to ensure all points are valid.

Step-by-Step Review:


Step 1: Solving the inequality

The inequality given is n+5n5>2\frac{n+5}{n-5} > 2. Multiplying through by n5n - 5 (assuming n>5n > 5 to keep the sign consistent), you get:

n+5>2(n5)n+5 > 2(n-5)

Simplify the right-hand side:

n+5>2n10n + 5 > 2n - 10

Rearrange terms:

15>n15 > n

or equivalently:

n<15n < 15

This step is correct, and n<15n < 15 is valid.


Step 2: Ensuring n+5n+5 is a twin prime

A twin prime pp satisfies that either p2p-2 or p+2p+2 is also prime. Listing twin primes up to n+5=19n+5 = 19:

  • (3,5)(3, 5), (5,7)(5, 7), (11,13)(11, 13), (17,19)(17, 19)

From this list, n+5n+5 values matching the inequality n<15n < 15 are:

  • n+5=5n+5 = 5 leads to n=0n = 0
  • n+5=7n+5 = 7 leads to n=2n = 2
  • n+5=11n+5 = 11 leads to n=6n = 6
  • n+5=13n+5 = 13 leads to n=8n = 8
  • n+5=17n+5 = 17 leads to n=12n = 12
  • n+5=19n+5 = 19 leads to n=14n = 14

This step is also correct.


Step 3: Ensuring n+8n+8 is not a prime number

We now test n+8n+8 for each valid nn from the list:

  • n=0n = 0: n+8=8n+8 = 8 (not prime)
  • n=2n = 2: n+8=10n+8 = 10 (not prime)
  • n=6n = 6: n+8=14n+8 = 14 (not prime)
  • n=8n = 8: n+8=16n+8 = 16 (not prime)
  • n=12n = 12: n+8=20n+8 = 20 (not prime)
  • n=14n = 14: n+8=22n+8 = 22 (not prime)

All values satisfy n+8n+8 not being a prime number. The largest nn satisfying all constraints is n=14n = 14.


Final Conclusion:

The solution is correct, and the maximum value of nn is indeed:

14\boxed{14}

Questions to Expand Understanding:

  1. How would the solution change if the inequality were n+5n52\frac{n+5}{n-5} \geq 2?
  2. Can you prove why n+8n+8 is never prime for all n<15n < 15 satisfying n+5n+5 being a twin prime?
  3. What happens if we extend the condition to include higher twin primes (e.g., beyond n+5=19n+5 = 19)?
  4. Is there a general approach to solving similar problems with twin primes?
  5. Can you use modular arithmetic to systematically eliminate values of nn?

Tip:

When dealing with primes or twin primes, it’s helpful to use modular arithmetic or sieve methods to verify conditions efficiently.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Prime Numbers
Twin Primes

Formulas

(n+5)/(n−5) > 2
n+5 > 2(n−5)

Theorems

Definition of Twin Primes
Properties of Prime Numbers

Suitable Grade Level

Grades 10-12