2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + 132020 + 132021 + 172022
2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + 132020 + 132021 + 172022
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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![### Mathematics Problem Set
1. **Use Euler’s Theorem**
Prove \( a^{265} \equiv a \pmod{105} \) for all \( a \in \mathbb{Z} \).
2. **Use Fermat’s Little Theorem**
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**
Use Wilson’s Theorem to prove \( (6(k - 4))! \equiv 1 \pmod{k} \), if \( k \) is prime.
4. **Factorization Using Fermat’s Method**
Use Fermat’s factorization method to factor 2168495737.
5. **Factorization Using Kraitchik’s Method**
Use Kraitchik’s factorization method to factor 11653.
6. **Prove the Following Function**
Prove \( \phi(k^2) = k \cdot \phi(k) \) for all \( k \in \mathbb{N} \).
7. **Proofs Based on Number Properties**
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 Modulo Proof**
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%2Fc575f1af-299e-48ac-b19b-82d3ce77357f%2Flz3zo9v.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematics Problem Set
1. **Use Euler’s Theorem**
Prove \( a^{265} \equiv a \pmod{105} \) for all \( a \in \mathbb{Z} \).
2. **Use Fermat’s Little Theorem**
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**
Use Wilson’s Theorem to prove \( (6(k - 4))! \equiv 1 \pmod{k} \), if \( k \) is prime.
4. **Factorization Using Fermat’s Method**
Use Fermat’s factorization method to factor 2168495737.
5. **Factorization Using Kraitchik’s Method**
Use Kraitchik’s factorization method to factor 11653.
6. **Prove the Following Function**
Prove \( \phi(k^2) = k \cdot \phi(k) \) for all \( k \in \mathbb{N} \).
7. **Proofs Based on Number Properties**
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 Modulo Proof**
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 \( k \).
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