(2) Consider the prime number p = 31 (a) Prove that gcd (7,31) 7x + 31y = 1. Prove that gcd(7, 30) = 1 by computing integers u and v such that 7u+ 30v 1. = 1 by computing integers x and y such that
(2) Consider the prime number p = 31 (a) Prove that gcd (7,31) 7x + 31y = 1. Prove that gcd(7, 30) = 1 by computing integers u and v such that 7u+ 30v 1. = 1 by computing integers x and y such that
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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![### Number Theory Problem Set
#### Problem 2: Prime Number \( p = 31 \)
##### (a) Prove that \(\gcd(7, 31) = 1\)
- **Objective:** Compute integers \(x\) and \(y\) such that:
\[
7x + 31y = 1
\]
- **Objective:** Compute integers \(u\) and \(v\) such that:
\[
7u + 30v = 1
\]
##### (b) Compute the smallest positive integer \(d\)
- **Given:** \(e = 7\)
- **Objective:** Find the smallest positive integer \(d\) such that:
\[
de \equiv 1 \pmod{p-1}
\]
Equivalently,
\[
7d \equiv 1 \pmod{30}
\]
##### (c) Compute the smallest positive integer \(x\)
- **Objective:** Find the smallest positive integer \(x\) such that:
\[
x^7 \equiv 24 \pmod{31}
\]
##### (d) Determine the primitiveness of roots modulo 31
- **Objective:** Prove the following:
- \(g = 2\) is not a primitive root modulo 31.
- \(g = 3\) is a primitive root modulo 31.
- Determine if \(g = 5\) is a primitive root modulo 31.
##### (e) Compute discrete logarithms
- **Objective:** For \(p = 31\), compute the discrete logarithms:
\[
\log_3{(17)} \quad \text{and} \quad \log_3{(2)}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa332eac-d846-4704-9340-0a50b86bfcea%2F52ef6607-09ba-46e9-8f61-9f843344dedb%2Fs7yu8j_processed.png&w=3840&q=75)
Transcribed Image Text:### Number Theory Problem Set
#### Problem 2: Prime Number \( p = 31 \)
##### (a) Prove that \(\gcd(7, 31) = 1\)
- **Objective:** Compute integers \(x\) and \(y\) such that:
\[
7x + 31y = 1
\]
- **Objective:** Compute integers \(u\) and \(v\) such that:
\[
7u + 30v = 1
\]
##### (b) Compute the smallest positive integer \(d\)
- **Given:** \(e = 7\)
- **Objective:** Find the smallest positive integer \(d\) such that:
\[
de \equiv 1 \pmod{p-1}
\]
Equivalently,
\[
7d \equiv 1 \pmod{30}
\]
##### (c) Compute the smallest positive integer \(x\)
- **Objective:** Find the smallest positive integer \(x\) such that:
\[
x^7 \equiv 24 \pmod{31}
\]
##### (d) Determine the primitiveness of roots modulo 31
- **Objective:** Prove the following:
- \(g = 2\) is not a primitive root modulo 31.
- \(g = 3\) is a primitive root modulo 31.
- Determine if \(g = 5\) is a primitive root modulo 31.
##### (e) Compute discrete logarithms
- **Objective:** For \(p = 31\), compute the discrete logarithms:
\[
\log_3{(17)} \quad \text{and} \quad \log_3{(2)}
\]
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