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Prove the following for Integers a, b, c, d, and e, ab be bc a | d(e - c)

Prove the following for Integers a, b, c, d, and e, ab be bc a | d(e - c)

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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**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 \( d(e - c) \).
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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