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3. Suppose p = 2 (mod 5) and p is a prime number. Show that if x5 = 1 (mod p), then x = 1 (mod p).

3. Suppose p = 2 (mod 5) and p is a prime number. Show that if x5 = 1 (mod p), then x = 1 (mod p).

Advanced Engineering Mathematics
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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**Problem Statement:** 3. Suppose \( p \equiv 2 \ (\text{mod} \ 5) \) and \( p \) is a prime number. Show that if \( x^5 \equiv 1 \ (\text{mod} \ p) \), then \( x \equiv 1 \ (\text{mod} \ p) \). **Explanation:** This mathematical problem involves concepts from number theory, specifically dealing with modular arithmetic and prime numbers. - **p \equiv 2 \ (\text{mod} \ 5):** This means that when the prime number \( p \) is divided by 5, it leaves a remainder of 2. - **x^5 \equiv 1 \ (\text{mod} \ p):** This states that \( x^5 \) is congruent to 1 modulo \( p \). In simpler terms, when \( x^5 \) is divided by \( p \), it leaves a remainder of 1. - **Show that x \equiv 1 \ (\text{mod} \ p):** The task is to prove that under the given conditions, \( x \) must also be congruent to 1 modulo \( p \), meaning \( x \) when divided by \( p \) should yield a remainder of 1. This type of problem is usually solved using properties of prime numbers and modular arithmetic to demonstrate that the conditions lead to a specific mathematical result.
Transcribed Image Text:**Problem Statement:** 3. Suppose \( p \equiv 2 \ (\text{mod} \ 5) \) and \( p \) is a prime number. Show that if \( x^5 \equiv 1 \ (\text{mod} \ p) \), then \( x \equiv 1 \ (\text{mod} \ p) \). **Explanation:** This mathematical problem involves concepts from number theory, specifically dealing with modular arithmetic and prime numbers. - **p \equiv 2 \ (\text{mod} \ 5):** This means that when the prime number \( p \) is divided by 5, it leaves a remainder of 2. - **x^5 \equiv 1 \ (\text{mod} \ p):** This states that \( x^5 \) is congruent to 1 modulo \( p \). In simpler terms, when \( x^5 \) is divided by \( p \), it leaves a remainder of 1. - **Show that x \equiv 1 \ (\text{mod} \ p):** The task is to prove that under the given conditions, \( x \) must also be congruent to 1 modulo \( p \), meaning \( x \) when divided by \( p \) should yield a remainder of 1. This type of problem is usually solved using properties of prime numbers and modular arithmetic to demonstrate that the conditions lead to a specific mathematical result.
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