.6. Let a, b, c € Z. Use the definition of divisibility to directly prove the following roperties of divisibility. (This is Proposition 1.4.) [a) If a | b and b | c, then a | c. b) If a | b and b|a, then a = ±b. c) If a b and a | c, then a (b + c) and a | (b − c).
.6. Let a, b, c € Z. Use the definition of divisibility to directly prove the following roperties of divisibility. (This is Proposition 1.4.) [a) If a | b and b | c, then a | c. b) If a | b and b|a, then a = ±b. c) If a b and a | c, then a (b + c) and a | (b − 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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Transcribed Image Text:### Properties of Divisibility
#### Proposition 1.4:
**Let \(a, b, c \in \mathbb{Z}\). Use the definition of divisibility to directly prove the following properties of divisibility:**
(a) If \(a \mid b\) and \(b \mid c\), then \(a \mid c\).
(b) If \(a \mid b\) and \(b \mid a\), then \(a = \pm b\).
(c) If \(a \mid b\) and \(a \mid c\), then \(a \mid (b + c)\) and \(a \mid (b - c)\).
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