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(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

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...
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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: \[ \log_3{(17)} \quad \text{and} \quad \log_3{(2)} \]
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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