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Prove or disprove: If a = b(mod n²) then a = b(mod n)

Prove or disprove: If a = b(mod n²) then a = b(mod n)

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 or disprove the following statement: If \( a \equiv b \pmod{n^2} \) then \( a \equiv b \pmod{n} \). **Explanation**: This problem involves modular arithmetic. Here, we're asked to assess whether the congruence \( a \equiv b \pmod{n^2} \) implies \( a \equiv b \pmod{n} \). **Key Points**: - **Congruence Modulo \( n^2 \)**: When numbers \( a \) and \( b \) are congruent modulo \( n^2 \), it means that the difference \( a - b \) is divisible by \( n^2 \). - **Congruence Modulo \( n \)**: When numbers \( a \) and \( b \) are congruent modulo \( n \), it means that the difference \( a - b \) is divisible by \( n \). **Potential Steps**: 1. **Start with the given congruence**: - \( a \equiv b \pmod{n^2} \): This implies \( a - b = k \cdot n^2 \) for some integer \( k \). 2. **Examine the implication**: - Analyze whether dividing this relationship by \( n \) still retains integer divisibility. - Since \( k \cdot n^2 = (k \cdot n) \cdot n \), it's evident that \( a - b = m \cdot n \) where \( m = k \cdot n \). 3. **Conclusion**: - Therefore, \( a \equiv b \pmod{n} \). This statement is true and can be proven since if the difference of two numbers is divisible by a larger modulus \( n^2 \), it must also be divisible by the smaller modulus \( n \).
Transcribed Image Text:**Problem Statement**: Prove or disprove the following statement: If \( a \equiv b \pmod{n^2} \) then \( a \equiv b \pmod{n} \). **Explanation**: This problem involves modular arithmetic. Here, we're asked to assess whether the congruence \( a \equiv b \pmod{n^2} \) implies \( a \equiv b \pmod{n} \). **Key Points**: - **Congruence Modulo \( n^2 \)**: When numbers \( a \) and \( b \) are congruent modulo \( n^2 \), it means that the difference \( a - b \) is divisible by \( n^2 \). - **Congruence Modulo \( n \)**: When numbers \( a \) and \( b \) are congruent modulo \( n \), it means that the difference \( a - b \) is divisible by \( n \). **Potential Steps**: 1. **Start with the given congruence**: - \( a \equiv b \pmod{n^2} \): This implies \( a - b = k \cdot n^2 \) for some integer \( k \). 2. **Examine the implication**: - Analyze whether dividing this relationship by \( n \) still retains integer divisibility. - Since \( k \cdot n^2 = (k \cdot n) \cdot n \), it's evident that \( a - b = m \cdot n \) where \( m = k \cdot n \). 3. **Conclusion**: - Therefore, \( a \equiv b \pmod{n} \). This statement is true and can be proven since if the difference of two numbers is divisible by a larger modulus \( n^2 \), it must also be divisible by the smaller modulus \( n \).
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