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Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for each n E Z4 with n > 2, there is r e Z4 and primes p1, P2, -..., Pr such that П Pi = P1P2 · ·• Pr. n = i=1

Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for each n E Z4 with n > 2, there is r e Z4 and primes p1, P2, -..., Pr such that П Pi = P1P2 · ·• Pr. n = i=1

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 53EQ
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can someone please show all the work for this? i have no idea how to do this

 

We say that an integer n is prime if n ¢ {-1,0,1} and 6. for all a, b e Z, if n|ab, then n|a or n|b. Remark. We say that an integer n is irreducible if n ¢ {-1,0,1} and for all a, b E Z, if n = ab, then a ±1 or b =±E1. This is probably the definition you were given for prime in your grade school algebra class. It turns out that an integer is prime if and only if it is irreducible. However, the notions of irreducible and prime are not the same in other contexts you will encounter in Math 100/103. Prove Theorem I below. You may use the following fact without proof: for every positive integer n > 2, either n is prime, or there exist positive integers a, b such that ab and 1 < a <b<n. n = Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for each n E Z, with n > 2, there is r E Z4 and primes P1, P2, . .. , Pr such that п — || Pi = PiP2 Pr. i=1
Transcribed Image Text:We say that an integer n is prime if n ¢ {-1,0,1} and 6. for all a, b e Z, if n|ab, then n|a or n|b. Remark. We say that an integer n is irreducible if n ¢ {-1,0,1} and for all a, b E Z, if n = ab, then a ±1 or b =±E1. This is probably the definition you were given for prime in your grade school algebra class. It turns out that an integer is prime if and only if it is irreducible. However, the notions of irreducible and prime are not the same in other contexts you will encounter in Math 100/103. Prove Theorem I below. You may use the following fact without proof: for every positive integer n > 2, either n is prime, or there exist positive integers a, b such that ab and 1 < a <b<n. n = Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for each n E Z, with n > 2, there is r E Z4 and primes P1, P2, . .. , Pr such that п — || Pi = PiP2 Pr. i=1
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