1 use the euclidean algorithm to find gcd162 48 then express...

Name: 1 use the euclidean algorithm to find gcd162 48 then express the gcd as a linear combination of 162 and 48 5 pts 39592
Uploaded: 2022-06-13T08:36:44-08:00
Duration: 2 min 42 s
Channel: Adi S
Description: 1 use the euclidean algorithm to find gcd162 48 then express the gcd as a linear combination of 162 and 48 5 pts 39592

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EXAMPLE 17: Express gcd(252, 198) = 18 as a linear combination of 252 and 198 by working backwards through the steps of the Euclidean algorithm. Solution: To show that gcd(252, 198) = 18, the Euclidean algorithm uses these divisions: 252 = 198 * 1 + 54 198 = 54 * 3 + 36 54 = 36 * 1 + 18 36 = 18 * 2 + 0. We summarize these steps in tabular form: 252 | 198 | 36 | 3 198 | 54 | 18 | 2 Using the next-to-last division (the third division), we can express gcd(252, 198) = 18 as a linear combination of 54 and 36. We find that: 18 = 54 - 1 * 36. The second division tells us that: 36 = 198 - 3 * 54. Substituting this expression for 36 into the previous equation, we can express 18 as a linear combination of 54 and 198. We have: 18 = 54 - 1 * 36 = 54 - 1 * (198 - 3 * 54) = 4 * 54 - 1 * 198. The first division tells us that: 54 = 252 - 1 * 198. Substituting this expression for 54 into the previous equation, we can express 18 as a linear combination of 252 and 198. We conclude that: 18 = 4 * (252 - 1 * 198) - 1 * 198, completing the solution. The next example shows how to solve the same problem posed in the previous example using the extended Euclidean algorithm.

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Transcript

00:02 In this problem, first we have to express gcd of 252 and 198 is equals to 18 as a linear combination of 252 and 198 by working backwards.

00:17 Suppose let us assume that there exists a and b.

00:21 These are constants such that 252a plus 198 b is equal to 18.

00:31 Where a comma b belongs to integers.

00:36 Now by euclidean algorithm, we find that 252 is equals to 198 into 1 plus 54 and 198 is equals to 54 into 3 plus 36 also 54 is equals to 36 into 1 plus 18...

Question

1. Use the Euclidean Algorithm to find gcd(162, -48). Then express the gcd as a linear combination of 162 and -48. (5 pts)

Question

1. Use the Euclidean Algorithm to find gcd(162, -48). Then express the gcd as a linear combination of 162 and -48. (5 pts)

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