1 ## Transcription for Educational Website### Mathematical Problems and Theorems1. **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

To prove: a265≡amod 105 for all a∈ℤ by using Euler's theorem. Euler's theorem:…

Answered: 1. Use Euler's Theorem to prove a 265 =…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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