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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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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
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![**Problem 1.**
1. *What are the squares modulo 4?*
2. *Let \( p \) be an odd prime and assume that \( p = a^2 + b^2 \) for some \( a, b \in \mathbb{Z} \). Show that*
\[
p \equiv 1 \pmod{4}.
\]
3. *Show that if \( p \) is an odd prime satisfying \( p \equiv 3 \pmod{4} \), then \( p \) is irreducible in \( \mathbb{Z}[i] \).*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F342166a8-a59c-4a53-933f-8a63a55a0b29%2F6297f610-7c5b-4a46-ab37-86c106b0ac93%2Fn88mov5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1.**
1. *What are the squares modulo 4?*
2. *Let \( p \) be an odd prime and assume that \( p = a^2 + b^2 \) for some \( a, b \in \mathbb{Z} \). Show that*
\[
p \equiv 1 \pmod{4}.
\]
3. *Show that if \( p \) is an odd prime satisfying \( p \equiv 3 \pmod{4} \), then \( p \) is irreducible in \( \mathbb{Z}[i] \).*
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