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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
Section1.2: Exponents And Radicals
Problem 92E
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**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.
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