PRODD 2 Determine whether the integer 701 is divisors. Do the same for the integer 1 Sieve of
PRODD 2 Determine whether the integer 701 is divisors. Do the same for the integer 1 Sieve of
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.4: Prime Factors And Greatest Common Divisor
Problem 4E: Find the smallest integer in the given set.
{ and for some in }
{ and for some in }
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Question 1
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PROBLEMS 3.2
Determine whether the integer 701 is prime by testing all primes p < 701 as possible
divisors. Do the same for the integer 1009.
Employing the Sieve of Eratosthenes, obtain all the primes between 100 and 200.
Z Given that p X n for all primes p < In, show that n > 1 is either a prime or the product
of two primes.
[Hint: Assume to the contrary that n contains at least three prime factors.]
4 Establish the following facts:
(a) p is irrational for any prime p.
(b) If a is a positive integer and a is rational, then a must be an integer.
(c) For n > 2, /n is irrational.
[Hint: Use the fact that 2" > n.]
5. Show that any composite three-digit number must have a prime factor less than or equal
to 31.
6. Fill in any missing details in this sketch of a proof of the infinitude of primes: Assume
that there are only finitely many primes, say p1, P2, ..., Pn. Let A be the product of any
r of these primes and put B
not both. Because A + B > 1, A + B has a prime divisor different from any of the pk,
which is a contradiction.
= PiP2· Pn/A. Then each pk divides either A or B, but
1. Modify Euclid's proof that there are infinitely many primes by assuming the existence
Of a largest prime p and using the integer = p! + 1 to arrive at a contradiction.
0. Give another proof of the infinitude of primes by assuming that there are only finitely many
||
Primes, say pi, P2, . .. , pn, and using the following integer to arrive at a contradiction:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F5fc4c4b7-a040-4b8c-bdbc-fbd2bbdf8cc1%2Fbai8qqi.jpeg&w=3840&q=75)
Transcribed Image Text:upto
Squ
PROBLEMS 3.2
Determine whether the integer 701 is prime by testing all primes p < 701 as possible
divisors. Do the same for the integer 1009.
Employing the Sieve of Eratosthenes, obtain all the primes between 100 and 200.
Z Given that p X n for all primes p < In, show that n > 1 is either a prime or the product
of two primes.
[Hint: Assume to the contrary that n contains at least three prime factors.]
4 Establish the following facts:
(a) p is irrational for any prime p.
(b) If a is a positive integer and a is rational, then a must be an integer.
(c) For n > 2, /n is irrational.
[Hint: Use the fact that 2" > n.]
5. Show that any composite three-digit number must have a prime factor less than or equal
to 31.
6. Fill in any missing details in this sketch of a proof of the infinitude of primes: Assume
that there are only finitely many primes, say p1, P2, ..., Pn. Let A be the product of any
r of these primes and put B
not both. Because A + B > 1, A + B has a prime divisor different from any of the pk,
which is a contradiction.
= PiP2· Pn/A. Then each pk divides either A or B, but
1. Modify Euclid's proof that there are infinitely many primes by assuming the existence
Of a largest prime p and using the integer = p! + 1 to arrive at a contradiction.
0. Give another proof of the infinitude of primes by assuming that there are only finitely many
||
Primes, say pi, P2, . .. , pn, and using the following integer to arrive at a contradiction:
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