Use the Euclidean Algorithm to a. Find the GCD of (5610, 8778) b. Express the GCD as a linear combination of the given numbers.
Use the Euclidean Algorithm to a. Find the GCD of (5610, 8778) b. Express the GCD as a linear combination of the given numbers.
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
Use the definition and example 1 as a reference to answer the question
Transcribed Image Text:Use the Euclidean Algorithm to
a. Find the GCD of (5610, 8778)
b. Express the GCD as a linear combination of the given numbers.
![Euclidean Algorithm. Given two integers a and b, where a > 0. In the
division algorithm,
b=aq₁ + ₁
0sr₂ <a
05r₂ <1₁
a=r₁ q1 + 7₂
7₁ = 7₂ 91 + 13
0573 <72
Tk -1 = Tk qk+1
Then T = (a, b)
Example 1. Find (1081, 1363)
1363=1 (1081) +282
1081=3 (282) +235
2821 (235) +47
235-5(47)
Thus, (1363, 1081)=47
Linear Form of the Greatest Common Divisor:
If d = (a, b), then d is the least natural number of the form ax + by.
Example 1. Find (1081, 1363)
47=282-1 (235)
235=1081-3 (282) and
282=1363-1 (1 081)
Thus, by substitution, we have
47 = 282-1 (235)
= 282 - [1 081-3 (282)]
=
4 (282) - 1081
= 4 (1 363-1081) - (1 081)
Finally, 47=4 (1363) - 5 (1 081)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfe06831-bdf4-41df-a46d-33d93803c01c%2F583e3a13-9ae3-457a-b334-e388dcf10de9%2Fq8pkf8u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Euclidean Algorithm. Given two integers a and b, where a > 0. In the
division algorithm,
b=aq₁ + ₁
0sr₂ <a
05r₂ <1₁
a=r₁ q1 + 7₂
7₁ = 7₂ 91 + 13
0573 <72
Tk -1 = Tk qk+1
Then T = (a, b)
Example 1. Find (1081, 1363)
1363=1 (1081) +282
1081=3 (282) +235
2821 (235) +47
235-5(47)
Thus, (1363, 1081)=47
Linear Form of the Greatest Common Divisor:
If d = (a, b), then d is the least natural number of the form ax + by.
Example 1. Find (1081, 1363)
47=282-1 (235)
235=1081-3 (282) and
282=1363-1 (1 081)
Thus, by substitution, we have
47 = 282-1 (235)
= 282 - [1 081-3 (282)]
=
4 (282) - 1081
= 4 (1 363-1081) - (1 081)
Finally, 47=4 (1363) - 5 (1 081)
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 2 steps with 2 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,
