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20. If a € Z and a = 1 (mod 5), then a² = 1 (mod 5). 23. Let a, b, c € Z and n EN. If a = b (mod n), then ca = cb (mod n).

20. If a € Z and a = 1 (mod 5), then a² = 1 (mod 5). 23. Let a, b, c € Z and n EN. If a = b (mod n), then ca = cb (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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Certainly! Here's the transcription and description for an educational website: --- ### Transcription of Educational Text **20.** If \( a \in \mathbb{Z} \) and \( a \equiv 1 \pmod{5} \), then \( a^2 \equiv 1 \pmod{5} \). *Redacted Text* **23.** Let \( a, b, c \in \mathbb{Z} \) and \( n \in \mathbb{N} \). If \( a \equiv b \pmod{n} \), then \( ca \equiv cb \pmod{n} \). ### Explanation **Concepts Covered:** - **Modular Arithmetic:** The text provides insights into properties of congruences in modular arithmetic. This is an important area in number theory and is used widely in modern applications such as cryptography. - **Basic Theorems:** - Theorem from line 20 addresses the property where if an integer \( a \) is congruent to 1 modulo 5, then its square will also be congruent to 1 modulo 5. - Theorem from line 23 showcases the property of multiplying both sides of a modular congruence by another integer, preserving equality if multiplication is performed consistently on both sides. These are foundational rules that help in solving more complex modular equations and understanding number properties under modular systems.
Transcribed Image Text:Certainly! Here's the transcription and description for an educational website: --- ### Transcription of Educational Text **20.** If \( a \in \mathbb{Z} \) and \( a \equiv 1 \pmod{5} \), then \( a^2 \equiv 1 \pmod{5} \). *Redacted Text* **23.** Let \( a, b, c \in \mathbb{Z} \) and \( n \in \mathbb{N} \). If \( a \equiv b \pmod{n} \), then \( ca \equiv cb \pmod{n} \). ### Explanation **Concepts Covered:** - **Modular Arithmetic:** The text provides insights into properties of congruences in modular arithmetic. This is an important area in number theory and is used widely in modern applications such as cryptography. - **Basic Theorems:** - Theorem from line 20 addresses the property where if an integer \( a \) is congruent to 1 modulo 5, then its square will also be congruent to 1 modulo 5. - Theorem from line 23 showcases the property of multiplying both sides of a modular congruence by another integer, preserving equality if multiplication is performed consistently on both sides. These are foundational rules that help in solving more complex modular equations and understanding number properties under modular systems.
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