3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.
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 and Explanation of Mathematical Problems
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:**
\[ 6(k-4)! \equiv 1 \pmod{k}, \]
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 \( a \) 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 \( k \).
This document presents several mathematical problems and theorems, including Euler's Theorem, Fermat's Little Theorem, Wilson's Theorem, and factorization methods, along with exercises about number theory concepts such as order, primitive roots, and the Euler totient function \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F55b4b857-0d92-4ecd-abd0-be293d523383%2Fdjup9p8.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription and Explanation of Mathematical Problems
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:**
\[ 6(k-4)! \equiv 1 \pmod{k}, \]
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 \( a \) 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 \( k \).
This document presents several mathematical problems and theorems, including Euler's Theorem, Fermat's Little Theorem, Wilson's Theorem, and factorization methods, along with exercises about number theory concepts such as order, primitive roots, and the Euler totient function \
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