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 . Multiplying through by (assuming to keep the sign consistent), you get:
Simplify the right-hand side:
Rearrange terms:
or equivalently:
This step is correct, and is valid.
Step 2: Ensuring is a twin prime
A twin prime satisfies that either or is also prime. Listing twin primes up to :
- , , ,
From this list, values matching the inequality are:
- leads to
- leads to
- leads to
- leads to
- leads to
- leads to
This step is also correct.
Step 3: Ensuring is not a prime number
We now test for each valid from the list:
- : (not prime)
- : (not prime)
- : (not prime)
- : (not prime)
- : (not prime)
- : (not prime)
All values satisfy not being a prime number. The largest satisfying all constraints is .
Final Conclusion:
The solution is correct, and the maximum value of is indeed:
Questions to Expand Understanding:
- How would the solution change if the inequality were ?
- Can you prove why is never prime for all satisfying being a twin prime?
- What happens if we extend the condition to include higher twin primes (e.g., beyond )?
- Is there a general approach to solving similar problems with twin primes?
- Can you use modular arithmetic to systematically eliminate values of ?
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