7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0 (mod n) ... Is the statement true if n is even but not divisible by 4?
7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0 (mod n) ... Is the statement true if n is even but not divisible by 4?
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
![**Mathematics Exercise: Prime Numbers and Congruences**
1. **Prime Number of the Form \(1 + 4n^4\)**
- Task: Determine the prime number of the form \(1 + 4n^4\) and prove it's the only one.
- Hint: Research Sophie Germain's Identity to factor \(1 + 4n^4\).
2. **Congruence Proof for Prime Numbers \(p > 5\)**
- Task: Let p be a prime number such that \(p > 5\). Prove that \(p^2 - 1 \equiv 0 \pmod{24}\).
3. **Congruence Verification for Prime Number \(q\) and Integer \(n\)**
- Task: Let \(q\) be a prime and \(n \in \mathbb{N}\) such that \(1 \leq n < q\). Prove that \(q \mid \binom{q}{n}\).
4. **Theory of Congruences Verification**
- Task: Use the theory of congruences to verify that:
\[
25 \mid (2^{n+4} + 3^{n+2} - 5^{n+6})
\]
for all \(n \in \mathbb{N}\).
5. **Linear Congruence Solutions**
- Task: Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[
8x + 9y \equiv 10 \pmod{11}
\]
6. **Final Digit of a Sixth Power**
- Task: Determine the possibilities for the final digit of a sixth power of an integer.
7. **Sum of Cubes and Divisibility**
- Task: Prove that if \(n\) is an odd positive integer or divisible by 4, then:
\[
1^3 + 2^3 + 3^3 + \ldots + (n-1)^3 \equiv 0 \pmod{n}
\]
- Question: Is the statement true if \(n\) is even but not divisible by 4?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F5c5f51ad-a631-4a9b-9c78-71d194746812%2Fo1v0n3n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Exercise: Prime Numbers and Congruences**
1. **Prime Number of the Form \(1 + 4n^4\)**
- Task: Determine the prime number of the form \(1 + 4n^4\) and prove it's the only one.
- Hint: Research Sophie Germain's Identity to factor \(1 + 4n^4\).
2. **Congruence Proof for Prime Numbers \(p > 5\)**
- Task: Let p be a prime number such that \(p > 5\). Prove that \(p^2 - 1 \equiv 0 \pmod{24}\).
3. **Congruence Verification for Prime Number \(q\) and Integer \(n\)**
- Task: Let \(q\) be a prime and \(n \in \mathbb{N}\) such that \(1 \leq n < q\). Prove that \(q \mid \binom{q}{n}\).
4. **Theory of Congruences Verification**
- Task: Use the theory of congruences to verify that:
\[
25 \mid (2^{n+4} + 3^{n+2} - 5^{n+6})
\]
for all \(n \in \mathbb{N}\).
5. **Linear Congruence Solutions**
- Task: Using congruence theory (not brute force), find all solutions to the following linear congruence:
\[
8x + 9y \equiv 10 \pmod{11}
\]
6. **Final Digit of a Sixth Power**
- Task: Determine the possibilities for the final digit of a sixth power of an integer.
7. **Sum of Cubes and Divisibility**
- Task: Prove that if \(n\) is an odd positive integer or divisible by 4, then:
\[
1^3 + 2^3 + 3^3 + \ldots + (n-1)^3 \equiv 0 \pmod{n}
\]
- Question: Is the statement true if \(n\) is even but not divisible by 4?
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