19. Obtain the eight incongruent solutions (a) 5x+3y = 1 (mod 7) 3x + 2y = 4 (mod 7). (b) 7x + 3y = 6 (mod 11) 4x + 2y = 9 (mod 11). (c) 11x + 5y = 7 (mod 20) 6x + 3y = 8 (mod 20).
19. Obtain the eight incongruent solutions (a) 5x+3y = 1 (mod 7) 3x + 2y = 4 (mod 7). (b) 7x + 3y = 6 (mod 11) 4x + 2y = 9 (mod 11). (c) 11x + 5y = 7 (mod 20) 6x + 3y = 8 (mod 20).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
#20
![Title: Solving Systems of Linear Congruences
---
**Page 84 - Elementary Number Theory**
**Problem 19**: Obtain the eight incongruent solutions of the linear congruence \(3x + 4y \equiv 5 \pmod{7}\).
**Problem 20**: Find the solutions of each of the following systems of congruences:
- (a)
\[
\begin{aligned}
5x + 3y &\equiv 1 \pmod{7} \\
3x + 2y &\equiv 4 \pmod{7} \\
\end{aligned}
\]
- (b)
\[
\begin{aligned}
7x + 3y &\equiv 6 \pmod{11} \\
4x + 2y &\equiv 9 \pmod{11} \\
\end{aligned}
\]
- (c)
\[
\begin{aligned}
11x + 5y &\equiv 7 \pmod{20} \\
6x + 3y &\equiv 8 \pmod{20} \\
\end{aligned}
\]
---
**Explanation**:
In these problems, we are working on solving systems of linear congruences. Each system consists of two linear congruences that need to be solved simultaneously. The solutions provided will satisfy both congruences in each system under the given modulus.
The problems involve using the properties of congruences and often require methods such as substitution or the application of the Chinese Remainder Theorem to find the solutions within the specified modulus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F10fea2fa-9ae0-4ffa-a828-5d65d01e2004%2Fjzdn9g6.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Solving Systems of Linear Congruences
---
**Page 84 - Elementary Number Theory**
**Problem 19**: Obtain the eight incongruent solutions of the linear congruence \(3x + 4y \equiv 5 \pmod{7}\).
**Problem 20**: Find the solutions of each of the following systems of congruences:
- (a)
\[
\begin{aligned}
5x + 3y &\equiv 1 \pmod{7} \\
3x + 2y &\equiv 4 \pmod{7} \\
\end{aligned}
\]
- (b)
\[
\begin{aligned}
7x + 3y &\equiv 6 \pmod{11} \\
4x + 2y &\equiv 9 \pmod{11} \\
\end{aligned}
\]
- (c)
\[
\begin{aligned}
11x + 5y &\equiv 7 \pmod{20} \\
6x + 3y &\equiv 8 \pmod{20} \\
\end{aligned}
\]
---
**Explanation**:
In these problems, we are working on solving systems of linear congruences. Each system consists of two linear congruences that need to be solved simultaneously. The solutions provided will satisfy both congruences in each system under the given modulus.
The problems involve using the properties of congruences and often require methods such as substitution or the application of the Chinese Remainder Theorem to find the solutions within the specified modulus.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
