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6. For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) Let me N and let a, b, c € Z. If mfc and ca = cb (mod m), then a = b (mod m). (b) If m € N, then the only solutions of the equation X² = X in Z/mZ are Oz/mz and 1z/mz.

6. For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) Let me N and let a, b, c € Z. If mfc and ca = cb (mod m), then a = b (mod m). (b) If m € N, then the only solutions of the equation X² = X in Z/mZ are Oz/mz and 1z/mz.

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 6 ### Instructions: For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. ### Statements: #### (a) **Statement:** Let \( m \in \mathbb{N} \) and let \( a, b, c \in \mathbb{Z} \). If \( m \nmid c \) and \( ca \equiv cb \ (\text{mod}\ m) \), then \( a \equiv b \ (\text{mod}\ m) \). #### (b) **Statement:** If \( m \in \mathbb{N} \), then the only solutions of the equation \( X^2 = X \) in \( \mathbb{Z}/m\mathbb{Z} \) are \( 0_{\mathbb{Z}/m\mathbb{Z}} \) and \( 1_{\mathbb{Z}/m\mathbb{Z}} \).
Transcribed Image Text:## Problem 6 ### Instructions: For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. ### Statements: #### (a) **Statement:** Let \( m \in \mathbb{N} \) and let \( a, b, c \in \mathbb{Z} \). If \( m \nmid c \) and \( ca \equiv cb \ (\text{mod}\ m) \), then \( a \equiv b \ (\text{mod}\ m) \). #### (b) **Statement:** If \( m \in \mathbb{N} \), then the only solutions of the equation \( X^2 = X \) in \( \mathbb{Z}/m\mathbb{Z} \) are \( 0_{\mathbb{Z}/m\mathbb{Z}} \) and \( 1_{\mathbb{Z}/m\mathbb{Z}} \).
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