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For any integer k > 1, let p(k) be the product of all distinct prime numbers that are not greater than k. For example, p(2) = 2 and p(10) = 2*3*5*7 = 210. (1) n is a positive integer. If for every integer k > 1, none of the integers greater than 1 and less than or equal to k is a factor of p(k) + n, then what is the value of n? (2) Which of the following can be shown as a result of question (1) (Note: This is a single choice question.) a. There are only a finite number of primes. b. There is an infinite number of primes. Hint: Assuming there are only a finite number of primes, then what can you conclude from your answer to question (1) ?

For any integer k > 1, let p(k) be the product of all distinct prime numbers that are not greater than k. For example, p(2) = 2 and p(10) = 2*3*5*7 = 210. (1) n is a positive integer. If for every integer k > 1, none of the integers greater than 1 and less than or equal to k is a factor of p(k) + n, then what is the value of n? (2) Which of the following can be shown as a result of question (1) (Note: This is a single choice question.) a. There are only a finite number of primes. b. There is an infinite number of primes. Hint: Assuming there are only a finite number of primes, then what can you conclude from your answer to question (1) ?

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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For any integer k > 1, let p(k) be the product of all distinct prime numbers that are not greater than k. For example, p(2) = 2 and p(10) = 2*3*5*7 = 210. (1) n is a positive integer. If for every integer k > 1, none of the integers greater than 1 and less than or equal to k is a factor of p(k) + n, then what is the value of n? (2) Which of the following can be shown as a result of question (1) (Note: This is a single choice question.) a. There are only a finite number of primes. b. There is an infinite number of primes. Hint: Assuming there are only a finite number of primes, then what can you conclude from your answer to question (1) ?
Transcribed Image Text:For any integer k > 1, let p(k) be the product of all distinct prime numbers that are not greater than k. For example, p(2) = 2 and p(10) = 2*3*5*7 = 210. (1) n is a positive integer. If for every integer k > 1, none of the integers greater than 1 and less than or equal to k is a factor of p(k) + n, then what is the value of n? (2) Which of the following can be shown as a result of question (1) (Note: This is a single choice question.) a. There are only a finite number of primes. b. There is an infinite number of primes. Hint: Assuming there are only a finite number of primes, then what can you conclude from your answer to question (1) ?
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