Homework Help is Here – Start Your Trial Now!
Math
Calculus
2. If n is a prime number, it is possible that n! + 1 is not prime.

2. If n is a prime number, it is possible that n! + 1 is not prime.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.4: Fractional Expressions
Problem 70E
See similar textbooks
icon
Related questions
Question

TRUE or FALSE

DO NOT COPY FROM OTHER WEBSITES

Upvote guarenteed for a correct and detailed answer. Thank you!!!

**Concept Explanation:** 2. If \( n \) is a prime number, it is possible that \( n! + 1 \) is not prime. **Detailed Analysis:** - **Prime Numbers:** Numbers greater than 1 that have no divisors other than 1 and themselves. - **Factorial (n!):** The product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). **Understanding the Statement:** The statement discusses the nature of the expression \( n! + 1 \) when \( n \) is a prime number. The factorial of a number \( n \), especially when \( n \) is large, results in a large product. Adding 1 might intuitively seem like it could easily be prime, but this is not always the case. **Examples:** - For \( n = 5 \), \( n! = 120 \) and \( n! + 1 = 121 \), which is \( 11 \times 11 \), not a prime number. - For \( n = 7 \), \( n! = 5040 \) and \( n! + 1 = 5041 \), which is \( 71 \times 71 \), not a prime number. These examples illustrate that although \( n \) is a prime, \( n! + 1 \) may not be a prime. **Conclusion:** When working with such expressions involving factorials, it is essential to check for primality, as assumptions based on simpler scenarios can be misleading.
Transcribed Image Text:**Concept Explanation:** 2. If \( n \) is a prime number, it is possible that \( n! + 1 \) is not prime. **Detailed Analysis:** - **Prime Numbers:** Numbers greater than 1 that have no divisors other than 1 and themselves. - **Factorial (n!):** The product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). **Understanding the Statement:** The statement discusses the nature of the expression \( n! + 1 \) when \( n \) is a prime number. The factorial of a number \( n \), especially when \( n \) is large, results in a large product. Adding 1 might intuitively seem like it could easily be prime, but this is not always the case. **Examples:** - For \( n = 5 \), \( n! = 120 \) and \( n! + 1 = 121 \), which is \( 11 \times 11 \), not a prime number. - For \( n = 7 \), \( n! = 5040 \) and \( n! + 1 = 5041 \), which is \( 71 \times 71 \), not a prime number. These examples illustrate that although \( n \) is a prime, \( n! + 1 \) may not be a prime. **Conclusion:** When working with such expressions involving factorials, it is essential to check for primality, as assumptions based on simpler scenarios can be misleading.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Holt Mcdougal Larson Pre-algebra: Student Edition…
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax
SEE MORE TEXTBOOKS