7.3#11 **Euler's Generalization of Fermat's Theorem**1. **Theorem 5**: If $ m $ and $ n $ are relatively prime positive integers, prove that: \[ m^{\phi(n)} + n^{\phi(m)} \equiv 1 \pmod{mn} \]2. **Problem 6**: Fill in missing details in Euler’s theorem proof. Let $ p $ be a prime divisor of $ n $ and $\gcd(a, p) = 1$. By Fermat’s theorem: \[ a^{p-1} \equiv 1 \pmod{p} \] Therefore, \[ a^{p(p-1)} \equiv 1 \pmod{p^2} \] By induction, \[ a^{p^{k-1}(p-1)} \equiv 1 \pmod{p^k} \] For some integer $ t $, this becomes: \[ a^{\phi(n)} \equiv 1 \pmod{n} \]3. **Problem 7**: Find the units digit of $ 3^{100} $ using Euler’s theorem.4. **Problem 8**: - (a) Linear congruence problem. Show assenedtion for: \[ ax \equiv b \pmod{n} \] - (b) Solve: \[ 3x \equiv 5 \pmod{26}, \quad 13x \equiv 2 \pmod{40}, \quad 10x \equiv 21 \pmod{49} \]5. **Problem 9**: Use Euler’s theorem to evaluate $ 2^{1000000} \pmod{77} $.6. **Problem 10**: For any integer $ a $, show $ a $ and $ a_{n+1} $ have the last digit.7. **Problem 11**: Assertions for any prime $ p $: - (a) $ \tau(p!) = 2\tau((p-1)!) $ - (b) \( \sigma(p!) = (p+1)\sigma((p

Tau function : It is the number of divisors function, denoted by τ τ is a multiplicative…

Answered: For any prime p, establish each of the…

For any prime p, establish each of the assertions below: (a) t(p!) = 2t((p – 1)!). (b) o(p!) = (p + 1)o ((p – 1)!). (c) $(p!) = (p – 1)ø((p – 1)!). %3D %3D %3D |

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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