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1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

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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## Transcription for Educational Website ### Mathematical Problems and Theorems 1. **Use Euler’s Theorem:** Prove that \( a^{265} \equiv a \pmod{105} \) for all \( a \in \mathbb{Z} \). 2. **Use Fermat’s Little Theorem:** Find the units digit of \( 7^{2018} + 11^{2019} + 13^{2020} + 13^{2021} + 17^{2022} \). 3. **Use Wilson’s Theorem:** Prove \( (6(k-4)!) \equiv 1 \pmod{k} \), if \( k \) is prime. 4. **Fermat’s Factorization Method:** Factor the number 2168495737. 5. **Kraitchik’s Factorization Method:** Factor the number 11653. 6. **Prove for Euler’s Totient Function:** Prove \( \phi(k^2) = k \cdot \phi(k) \) for all \( k \in \mathbb{N} \). 7. **Prove Each of the Following Statements:** a) If \( q \) is a prime number not equal to 3 and \( k = 3q \), then \( \sigma(k) = 2 (\tau(k) + \phi(k)) \). b) If \( q \) is an odd prime number and \( k = 2q \), then \( k = \sigma(k) - \tau(k) - \phi(k) \). 8. **Inverse and Order of a Modulo:** Let \( \hat{a} \) be the inverse of \( a \) modulo \( k \). Prove that the order of \( a \) modulo \( k \) is equal to the order of \( \hat{a} \) modulo \( k \). Use this result to easily show that if \( a \) is a primitive root modulo \( k \) then \( \hat{a} \) is also a primitive root modulo \( k \). --- ### Explanation of Theoretical Concepts: - **Euler’s Theorem:** A generalization of Fermat's Little Theorem; it states that if two numbers \( a \) and \( n \) are copr
Transcribed Image Text:## Transcription for Educational Website ### Mathematical Problems and Theorems 1. **Use Euler’s Theorem:** Prove that \( a^{265} \equiv a \pmod{105} \) for all \( a \in \mathbb{Z} \). 2. **Use Fermat’s Little Theorem:** Find the units digit of \( 7^{2018} + 11^{2019} + 13^{2020} + 13^{2021} + 17^{2022} \). 3. **Use Wilson’s Theorem:** Prove \( (6(k-4)!) \equiv 1 \pmod{k} \), if \( k \) is prime. 4. **Fermat’s Factorization Method:** Factor the number 2168495737. 5. **Kraitchik’s Factorization Method:** Factor the number 11653. 6. **Prove for Euler’s Totient Function:** Prove \( \phi(k^2) = k \cdot \phi(k) \) for all \( k \in \mathbb{N} \). 7. **Prove Each of the Following Statements:** a) If \( q \) is a prime number not equal to 3 and \( k = 3q \), then \( \sigma(k) = 2 (\tau(k) + \phi(k)) \). b) If \( q \) is an odd prime number and \( k = 2q \), then \( k = \sigma(k) - \tau(k) - \phi(k) \). 8. **Inverse and Order of a Modulo:** Let \( \hat{a} \) be the inverse of \( a \) modulo \( k \). Prove that the order of \( a \) modulo \( k \) is equal to the order of \( \hat{a} \) modulo \( k \). Use this result to easily show that if \( a \) is a primitive root modulo \( k \) then \( \hat{a} \) is also a primitive root modulo \( k \). --- ### Explanation of Theoretical Concepts: - **Euler’s Theorem:** A generalization of Fermat's Little Theorem; it states that if two numbers \( a \) and \( n \) are copr
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