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
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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can someone please show all the work for this? i have no idea how to do this
![**Prime and Irreducible Numbers**
We say that an integer \( n \) is **prime** if \( n \notin \{-1, 0, 1\} \) and for all \( a, b \in \mathbb{Z} \), if \( n \mid ab \), then \( n \mid a \) or \( n \mid b \).
**Remark:** An integer \( n \) is **irreducible** if \( n \notin \{-1, 0, 1\} \) and for all \( a, b \in \mathbb{Z} \), if \( n = ab \), then \( a = \pm 1 \) or \( b = \pm 1 \).
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.
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**Prove Theorem 1 below.** You may use the following fact without proof: for every positive integer \( n \geq 2 \), either \( n \) is prime, or there exist positive integers \( a, b \) such that \( n = ab \) and \( 1 < a \leq b < n \).
**Theorem 1:** Every positive integer \( n \geq 2 \) can be written as a product of primes. In other words, for each \( n \in \mathbb{Z}_+ \) with \( n \geq 2 \), there is \( r \in \mathbb{Z}_+ \) and primes \( p_1, p_2, \ldots, p_r \) such that
\[ n = \prod_{i=1}^r p_i = p_1 p_2 \cdots p_r. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba208968-0ccc-4a94-8921-5ea260cefe19%2F127d215e-7ede-46e6-9095-3768b7a9adac%2Fbjt8hpk_processed.png&w=3840&q=75)
Transcribed Image Text:**Prime and Irreducible Numbers**
We say that an integer \( n \) is **prime** if \( n \notin \{-1, 0, 1\} \) and for all \( a, b \in \mathbb{Z} \), if \( n \mid ab \), then \( n \mid a \) or \( n \mid b \).
**Remark:** An integer \( n \) is **irreducible** if \( n \notin \{-1, 0, 1\} \) and for all \( a, b \in \mathbb{Z} \), if \( n = ab \), then \( a = \pm 1 \) or \( b = \pm 1 \).
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 1 below.** You may use the following fact without proof: for every positive integer \( n \geq 2 \), either \( n \) is prime, or there exist positive integers \( a, b \) such that \( n = ab \) and \( 1 < a \leq b < n \).
**Theorem 1:** Every positive integer \( n \geq 2 \) can be written as a product of primes. In other words, for each \( n \in \mathbb{Z}_+ \) with \( n \geq 2 \), there is \( r \in \mathbb{Z}_+ \) and primes \( p_1, p_2, \ldots, p_r \) such that
\[ n = \prod_{i=1}^r p_i = p_1 p_2 \cdots p_r. \]
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