The extended Euclidean algorithm computesthe gcd of two integers ro and r₁ as a linearcombination of the inputs.gcd(ro, r₁) = s.ro + t •rıHere s and t are integers known as the Bezoutcoefficients. They are not unique.The algorithm works like the standard Euclideanalgorithm, except that at each stage the currentremainderri is expressed as a linear combination of theinputs.ri = siro + tir₁.This produces a sequence of numbersTo, T1,Tn-1, Tn where r = 0 andgcd(ro, r₁) = Tn-1. Suppose that ro = 420 andT1 = 382.Give the sequence ro, T1,..., Tn-1, în in the blankbelow. Enter your answer as a comma separated list ofnumbers.What is GCD(420,382)?What is s?What is t?

Given Information:The numbers are given as and .Objective:Find the of and by Euclidean Algorithm.…

The extended Euclidean algorithm computes the gcd of two integers ro and r₁ as a linear combination of the inputs. gcd(ro, r₁) = Here s and t are integers known as the Bezout coefficients. They are not unique. = s · ro + t · r₁ The algorithm works like the standard Euclidean algorithm, except that at each stage the current remainder î¿ is expressed as a linear combination of the inputs. rį = sįro + t¿r₁. = This produces a sequence of numbers To, 1,,Tn-1, n where n 0 and gcd(ro, r₁) = Tn-1. Suppose that ro T₁ = 382. What is GCD(420,382)? What is s? Give the sequence ro, T1,..., Tn-1, în in the blank below. Enter your answer as a comma separated list of numbers. What is t? = 420 and

The extended Euclidean algorithm computes the gcd of two integers ro and r₁ as a linear combination of the inputs. gcd(ro, r₁) = Here s and t are integers known as the Bezout coefficients. They are not unique. = s · ro + t · r₁ The algorithm works like the standard Euclidean algorithm, except that at each stage the current remainder î¿ is expressed as a linear combination of the inputs. rį = sįro + t¿r₁. = This produces a sequence of numbers To, 1,,Tn-1, n where n 0 and gcd(ro, r₁) = Tn-1. Suppose that ro T₁ = 382. What is GCD(420,382)? What is s? Give the sequence ro, T1,..., Tn-1, în in the blank below. Enter your answer as a comma separated list of numbers. What is t? = 420 and

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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Question

The extended Euclidean algorithm computes
the gcd of two integers ro and r₁ as a linear
combination of the inputs.
gcd(ro, r₁) = s.ro + t •rı
Here s and t are integers known as the Bezout
coefficients. They are not unique.
The algorithm works like the standard Euclidean
algorithm, except that at each stage the current
remainder
ri is expressed as a linear combination of the
inputs.
ri = siro + tir₁.
This produces a sequence of numbers
To, T1,
Tn-1, Tn where r = 0 and
gcd(ro, r₁) = Tn-1. Suppose that ro = 420 and
T1 = 382.
Give the sequence ro, T1,..., Tn-1, în in the blank
below. Enter your answer as a comma separated list of
numbers.
What is GCD(420,382)?
What is s?
What is t?

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1. For each pair of integers m and n, use the division algorithm to divide m by n. Write the result in the form m = q⋅n + r where 0 < r < n. a. m = 32 and n = 9 b. m 62 and n = 7 = c. m -45 and n = 11 d. m 3 and n = 8 2. For each pair of integers a and b, use the Euclidean Algorithm to find the GCD(a, b). Then use Theorem 92 to find the LCM(a, b) a. a = 48 and b = 54 b. a = 330 and b = 156 c. a=1188 and b = 385 3. For each pair of integers m and d, calculate m mod d. Show your work! a. m 43 and d = 9 b. m 50 and d = 7 c. m 28 and d = 5 d. m = 30 and d = 2