Math Problem Statement
Let ( n ) be a positive integer such that ( n + 5 ) is a twin prime and ( n + 8 ) is not a prime number. Find the maximum possible value of ( n ) such that ( \frac{n+5}{n-5} > 2 ).
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Solution
We are given that is a twin prime, meaning it is part of a pair of prime numbers of the form , and that is not prime. Additionally, we are asked to find the maximum possible value of such that:
Step 1: Analyze the inequality
We start by simplifying the given inequality:
Multiply both sides by (assuming ):
Expand both sides:
Now, subtract from both sides:
Add 10 to both sides:
Thus, must be less than 15. We will now examine the conditions on .
Step 2: Conditions on twin primes
Since is a twin prime, it must be part of a pair of primes . We list all twin primes less than 15:
- and (so , giving , but must be positive)
- and (so , giving )
- and (so , giving )
- and (so , giving )
Thus, the possible values of are .
Step 3: Check the condition on
We are also given that is not a prime number. We check for each possible :
- If , then , which is not prime.
- If , then , which is not prime.
- If , then , which is not prime.
So, all these values of satisfy the condition that is not prime.
Step 4: Maximize
Since must be less than 15 and the maximum possible value of is 14, the maximum possible value of that satisfies all the conditions is:
Further Exploration:
- Can you find any other twin primes for values of greater than 15?
- How would the inequality change if we had instead?
- What happens if is a prime number? How would the conditions change?
- What is the relationship between twin primes and other types of prime pairs?
- How do you determine whether a number is a prime or not?
Tip: Always verify the boundary conditions and the problem constraints carefully to avoid overlooking potential solutions.
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Math Problem Analysis
Mathematical Concepts
Prime Numbers
Twin Primes
Inequalities
Formulas
\( \frac{n+5}{n-5} > 2 \)
Theorems
Definition of Prime Numbers
Definition of Twin Primes
Suitable Grade Level
Grades 9-12