2.5 (a) Prove that if x, y is a solution to ax+by = d, with d= gcd(a, b), then for all c Z, b x² = x + c • }, d' y = y-c₁ (2.6.1) is also a solution to ax + by = d. (b) Find two distinct solutions to 2261x + 1275y = 17. (c) Prove that all solutions are of the form (2.6.1) for some c.
2.5 (a) Prove that if x, y is a solution to ax+by = d, with d= gcd(a, b), then for all c Z, b x² = x + c • }, d' y = y-c₁ (2.6.1) is also a solution to ax + by = d. (b) Find two distinct solutions to 2261x + 1275y = 17. (c) Prove that all solutions are of the form (2.6.1) for some c.
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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![## Section 2.5
### (a) Proof of Solution Form
Prove that if \( x, y \) is a solution to the equation \( ax + by = d \), where \( d = \gcd(a, b) \), then for all \( c \in \mathbb{Z} \),
\[
x' = x + c \cdot \frac{b}{d}, \quad y' = y - c \cdot \frac{a}{d} \quad \text{(2.6.1)}
\]
is also a solution to \( ax + by = d \).
### (b) Finding Distinct Solutions
Find two distinct solutions to the equation \( 2261x + 1275y = 17 \).
### (c) Proving Solution Form
Prove that all solutions are of the form given in equation (2.6.1) for some \( c \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe87a569b-7721-4104-9b29-fedcf0a2817b%2Fe1141a04-42e3-48a3-947d-74a4c038742a%2Fk5f8d64_processed.png&w=3840&q=75)
Transcribed Image Text:## Section 2.5
### (a) Proof of Solution Form
Prove that if \( x, y \) is a solution to the equation \( ax + by = d \), where \( d = \gcd(a, b) \), then for all \( c \in \mathbb{Z} \),
\[
x' = x + c \cdot \frac{b}{d}, \quad y' = y - c \cdot \frac{a}{d} \quad \text{(2.6.1)}
\]
is also a solution to \( ax + by = d \).
### (b) Finding Distinct Solutions
Find two distinct solutions to the equation \( 2261x + 1275y = 17 \).
### (c) Proving Solution Form
Prove that all solutions are of the form given in equation (2.6.1) for some \( c \).
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