5. Use Kraitchik's factorization method to factor 11653.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
5
![**Mathematical Problems on Number Theory**
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 \( k \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2Fb99dca59-34db-4259-ba1b-aa996f5c0c6f%2Foolci0j.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Problems on Number Theory**
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 \( k \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
