Use the Euclidean Algorithm to a. Find the GCD of (5610, 8778) b. Express the GCD as a linear combination of the given numbers.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
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Chapter10: Sequences, Series, And Probability
Section10.5: The Binomial Theorem
Problem 13E
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Use the definition and example 1 as a reference to answer the question

Use the Euclidean Algorithm to
a. Find the GCD of (5610, 8778)
b. Express the GCD as a linear combination of the given numbers.
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)
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)
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