na ed t 4. Use the theory of congruences to verify that 25| (2"+4 + 33n+2 – 53n+6) for all n eN
na ed t 4. Use the theory of congruences to verify that 25| (2"+4 + 33n+2 – 53n+6) for all n eN
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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![Sure! Below is the transcription of the image suitable for an educational website:
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**Mathematics Problem Set**
1. Only one prime of the form \(1 + 4n^4\) exists. Determine this prime number and prove it’s the only one of this particular form. Hint: Research Sophie Germain’s Identity to factor \(1 + 4n^4\).
2. Let \( p \) be a prime number such that \( p > 5 \). Prove that \( p^2 - 1 \equiv 0 \pmod{24} \).
3. Let \( q \) be a prime and \( n \in \mathbb{N} \) such that \( 1 \leq n < q \). Prove that \( q \mid \binom{q}{n} \).
4. Use the theory of congruences to verify that:
\[
25 \mid \left( 2^{n+4} + 3^{n+2} - 5^{n+6} \right) \quad \text{for all } n \in \mathbb{N}
\]
5. Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[
8x + 9y \equiv 10 \pmod{11}
\]
6. Determine the possibilities for the final digit of a sixth power of an integer.
7. Prove that if \( n \) is an odd positive integer or divisible by 4, then:
\[
1^3 + 2^3 + 3^3 + \ldots + (n-1)^3 \equiv 0 \pmod{n}
\]
Is the statement true if \( n \) is even but not divisible by 4?
---
This problem set explores various aspects of number theory and congruences, challenging students to apply advanced mathematical concepts to solve these intriguing problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F99c8a903-4561-44a8-b8ba-104b9e6bedce%2Fbtgyd8e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Sure! Below is the transcription of the image suitable for an educational website:
---
**Mathematics Problem Set**
1. Only one prime of the form \(1 + 4n^4\) exists. Determine this prime number and prove it’s the only one of this particular form. Hint: Research Sophie Germain’s Identity to factor \(1 + 4n^4\).
2. Let \( p \) be a prime number such that \( p > 5 \). Prove that \( p^2 - 1 \equiv 0 \pmod{24} \).
3. Let \( q \) be a prime and \( n \in \mathbb{N} \) such that \( 1 \leq n < q \). Prove that \( q \mid \binom{q}{n} \).
4. Use the theory of congruences to verify that:
\[
25 \mid \left( 2^{n+4} + 3^{n+2} - 5^{n+6} \right) \quad \text{for all } n \in \mathbb{N}
\]
5. Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[
8x + 9y \equiv 10 \pmod{11}
\]
6. Determine the possibilities for the final digit of a sixth power of an integer.
7. Prove that if \( n \) is an odd positive integer or divisible by 4, then:
\[
1^3 + 2^3 + 3^3 + \ldots + (n-1)^3 \equiv 0 \pmod{n}
\]
Is the statement true if \( n \) is even but not divisible by 4?
---
This problem set explores various aspects of number theory and congruences, challenging students to apply advanced mathematical concepts to solve these intriguing problems.
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