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) ?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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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