Prove the following for Integers a, b, c, d, and e, ab be bc a | d(e - c)
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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![**Problem Statement:**
Prove the following for integers \( a, b, c, d, \) and \( e \):
- \( a \mid b \)
- \( b \mid e \)
- \( b \mid c \)
**Conclusion:**
Therefore, prove that:
\[ a \mid d(e - c) \]
**Explanation:**
The statement uses divisibility conditions to establish a relationship between integers. Here, \( a \mid b \) means \( a \) divides \( b \), which implies there exists an integer \( k \) such that \( b = ak \). Similarly, \( b \mid e \) and \( b \mid c \) imply that \( e = bm \) and \( c = bn \) for some integers \( m \) and \( n \), respectively. The task is to prove that \( a \) divides the product \( d(e - c) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F233b4b28-2634-4ce7-bb56-644fea0dac02%2F55b38671-c042-43dc-ac4c-dacfd43f4c12%2Fuslbwv8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Prove the following for integers \( a, b, c, d, \) and \( e \):
- \( a \mid b \)
- \( b \mid e \)
- \( b \mid c \)
**Conclusion:**
Therefore, prove that:
\[ a \mid d(e - c) \]
**Explanation:**
The statement uses divisibility conditions to establish a relationship between integers. Here, \( a \mid b \) means \( a \) divides \( b \), which implies there exists an integer \( k \) such that \( b = ak \). Similarly, \( b \mid e \) and \( b \mid c \) imply that \( e = bm \) and \( c = bn \) for some integers \( m \) and \( n \), respectively. The task is to prove that \( a \) divides the product \( d(e - c) \).
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