Run Euclid's extended GCD algorithm to compute gcd(576,486). Show all the steps.

Name: Problem 5, Chapter 2: Discrete Math in Computer Science
Uploaded: 2019-12-21T21:27:57-08:00
Duration: 8 min
Channel: Bryan Lynn
Description: Run Euclid's extended GCD algorithm to compute gcd(576,486). Show all the steps.

Run Euclid's extended GCD algorithm to compute gcd(576,486)....

Key Concepts

Greatest Common Divisor (gcd)

The greatest common divisor is the largest integer that divides two numbers without leaving a remainder. It is a fundamental concept in number theory and is used to simplify fractions, solve equations, and analyze integer relationships.

Euclidean Algorithm

The Euclidean algorithm is a method for finding the greatest common divisor of two integers. By repeatedly applying the division algorithm and replacing numbers by their remainders, this process efficiently reduces the problem until reaching a remainder of zero, at which point the last non-zero remainder is the gcd.

Extended Euclidean Algorithm

The extended Euclidean algorithm builds on the Euclidean algorithm by also finding integer coefficients that express the gcd as a linear combination of the original two numbers (Bézout's Identity). This is particularly useful in solving linear Diophantine equations and finding modular inverses in cryptography.

Bézout's Identity

Bézout's Identity states that the greatest common divisor of two integers can be expressed as a linear combination of those integers. This concept is central to the extended Euclidean algorithm and has applications in solving equations where integer solutions are required.

Transcript

00:01 All right, so here we're using the euclidean algorithm to find greatest common divisors.

00:05 And the greatest common divisor, the euclidean algorithm states that when we're finding the greatest common divisor, it's going to be the same as taking the greatest common divisor of the smaller of the two numbers and the remainder.

00:18 So let's use their examples to sort of walk through what that really looks like.

00:24 And so initially we're given to find the greatest common divisor of 12 and 18.

00:31 So this should be equal to taking the greatest common divisor of the smaller one, which is 12, with the remainder.

00:39 Right.

00:40 So when you divide 18 by 12, you get a remainder of 6.

00:46 Okay.

00:47 And then we can do the same thing.

00:49 So this should be the same as taking the greatest common divisor of 6.

00:53 And then we divide 6 by 12.

00:55 The remainder is 0.

00:58 Well, so now, now we just get this as equal to 6.

01:02 So we can't go down any further.

01:03 So let's try b.

01:06 Here we want the greatest common divisor of 111 and 201.

01:14 So this is going to be equal to taking the greatest common divisor.

01:18 We'll take the smaller one, 111, and the remainder when we divide this, right? so that would be 90.

01:28 And i'll do the same thing.

01:29 This will be equal to the greatest common divisor of take the smaller one 90 and the remainder, which is 21, which is going to be equal to taking the greatest common divisor.

01:45 We'll start with a smaller one, 21, and the remainder when we divide it into 90.

01:51 So 90 divided by 21 gives us a remainder of six.

01:57 So the same thing again, greatest common divisor.

02:00 We'll take the smaller one, six.

02:03 We'll divide it into 21 because this remainder of three.

02:10 Okay, so this is equal to taking the greatest common divisor, we'll take the smaller one, when we divide 6 by 3, we get 0.

02:18 So that gives us a 3 for our solution of the greatest common divisor.

02:25 So we'll go to a new page here, just in case this one is long.

02:31 And for part c, we want to find the greatest common divisor of 1 ,001, and 1331.

02:41 Okay, so we start by taking the smaller number, which in this case is 1001.

02:47 We would divide 1 ,331 by 1 ,001, and we'll keep the remainder, in this case is 330.

02:58 So now we'll take the greatest common divisor of 330.

03:03 We'll divide 1 ,001 by 330 and keep the remainder, which is 11.

03:11 So now we'll take the greatest common divisor of the smaller one, divide it 330 by 11, which i believe would just be 30 and 0...

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