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

1. Use Euler's Theorem to prove a = 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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1. Use Euler’s Theorem to prove \( a^{265} \equiv a \pmod{105} \) for all \( a \in \mathbb{Z} \). 2. Use Fermat’s Little Theorem, or its corollary, to find the units digit of \( 7^{2018} + 11^{2019} + 13^{2020} + 13^{2021} + 17^{2022} \). 3. Use Wilson’s Theorem to prove \( (6k-4)! \equiv 1 \pmod{k} \), if \( k \) is prime. 4. Use Fermat’s factorization method to factor 2168495737. 5. Use Kraitchik’s factorization method to factor 11653. 6. 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. Let \( \hat{a} \) be the inverse of \( a \pmod{k} \). Prove that the order of \( a \pmod{k} \) is equal to the order of \( \hat{a} \pmod{k} \). Use this result to easily show that if \( a \) is a primitive root modulo \( k \) then \( \hat{a} \) is also a primitive root modulo \( k \).
Transcribed Image Text:1. Use Euler’s Theorem to prove \( a^{265} \equiv a \pmod{105} \) for all \( a \in \mathbb{Z} \). 2. Use Fermat’s Little Theorem, or its corollary, to find the units digit of \( 7^{2018} + 11^{2019} + 13^{2020} + 13^{2021} + 17^{2022} \). 3. Use Wilson’s Theorem to prove \( (6k-4)! \equiv 1 \pmod{k} \), if \( k \) is prime. 4. Use Fermat’s factorization method to factor 2168495737. 5. Use Kraitchik’s factorization method to factor 11653. 6. 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. Let \( \hat{a} \) be the inverse of \( a \pmod{k} \). Prove that the order of \( a \pmod{k} \) is equal to the order of \( \hat{a} \pmod{k} \). Use this result to easily show that if \( a \) is a primitive root modulo \( k \) then \( \hat{a} \) is also a primitive root modulo \( k \).
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We need to prove a265≡a mod(105) for all a∈Z, by using Euler's theorem.

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