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 |
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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Question
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F323cd094-b780-4c09-9cd8-0dd7adccf3ba%2Fsj7tvsg.jpeg&w=3840&q=75)
Transcribed Image Text:**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
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