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5. Let p and q be distinct primes and let R = Z/pqZ. Prove that, if a ER satisfies a" = OR for some n € N, then a = OR. (Hint: Choose a representative a € Z for a. What does the equality a" = OR tell you about a? What would have to be true of a to have a = OR?)

5. Let p and q be distinct primes and let R = Z/pqZ. Prove that, if a ER satisfies a" = OR for some n € N, then a = OR. (Hint: Choose a representative a € Z for a. What does the equality a" = OR tell you about a? What would have to be true of a to have a = OR?)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 18EQ
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**Problem 5:** Let \( p \) and \( q \) be distinct primes and let \( R = \mathbb{Z}/pq\mathbb{Z} \). Prove that, if \( \alpha \in R \) satisfies \( \alpha^n = 0_R \) for some \( n \in \mathbb{N} \), then \( \alpha = 0_R \). *(Hint: Choose a representative \( a \in \mathbb{Z} \) for \( \alpha \). What does the equality \( \alpha^n = 0_R \) tell you about \( a \)? What would have to be true of \( a \) to have \( \alpha = 0_R \)?)*
Transcribed Image Text:**Problem 5:** Let \( p \) and \( q \) be distinct primes and let \( R = \mathbb{Z}/pq\mathbb{Z} \). Prove that, if \( \alpha \in R \) satisfies \( \alpha^n = 0_R \) for some \( n \in \mathbb{N} \), then \( \alpha = 0_R \). *(Hint: Choose a representative \( a \in \mathbb{Z} \) for \( \alpha \). What does the equality \( \alpha^n = 0_R \) tell you about \( a \)? What would have to be true of \( a \) to have \( \alpha = 0_R \)?)*
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