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Prove that if a E Z then the only positive divisor of both a and a + 1 is 1.

Prove that if a E Z then the only positive divisor of both a and a + 1 is 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Mathematical Proof Statement** **Problem Statement:** Prove that if \( a \in \mathbb{Z} \) then the only positive divisor of both \( a \) and \( a + 1 \) is 1. **Explanation:** This statement essentially claims that any integer \( a \) and its consecutive integer \( a + 1 \) are coprime, meaning that their greatest common divisor (gcd) is 1. To prove this, consider the following reasoning: - For any integer \( a \), suppose there is a common divisor \( d > 1 \) of both \( a \) and \( a + 1 \). - This implies that \( d \) divides both \( a \) and \( a + 1 - a = 1 \). - Since no integer greater than 1 can divide 1, the assumption that \( d > 1 \) is incorrect. - Therefore, the greatest common divisor of \( a \) and \( a + 1 \) is 1, proving they are coprime.
Transcribed Image Text:**Mathematical Proof Statement** **Problem Statement:** Prove that if \( a \in \mathbb{Z} \) then the only positive divisor of both \( a \) and \( a + 1 \) is 1. **Explanation:** This statement essentially claims that any integer \( a \) and its consecutive integer \( a + 1 \) are coprime, meaning that their greatest common divisor (gcd) is 1. To prove this, consider the following reasoning: - For any integer \( a \), suppose there is a common divisor \( d > 1 \) of both \( a \) and \( a + 1 \). - This implies that \( d \) divides both \( a \) and \( a + 1 - a = 1 \). - Since no integer greater than 1 can divide 1, the assumption that \( d > 1 \) is incorrect. - Therefore, the greatest common divisor of \( a \) and \( a + 1 \) is 1, proving they are coprime.
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