I'm taking a discrete math course, and were on Bézout Coefficients right now. I kind...

Luciano Webster

Luciano Webster

Answered

2022-07-17

I'm taking a discrete math course, and were on Bézout Coefficients right now. I kind of understand the algorithm, the generalization. However the example in the book is throwing me off.
The steps in the Euclidean algorithm to find gcd(101,4620) are:
4620 = 45 101 + 75 101 = 1 75 + 26 75 = 2 26 + 23 26 = 1 23 + 3 23 = 7 3 + 2 3 = 1 2 + 1 2 = 2 1
This I understand. Now to find the Bézout coefficients they follow these steps.
1 = 3 1 2 = 3 1 ( 23 7 3 ) = 1 23 + 8 3 = 1 23 + 8 ( 25 1 23 ) = 8 26 9 23 = 8 26 9 ( 75 2 26 ) = 0 75 + 26 26 = 0 75 + 26 ( 101 1 75 ) = 26 101 35 75 = 26 101 35 ( 4620 45 101 ) = 35 4620 + 1601 101
My problem is with the second line, where are they getting this +8 from? I've tried just about every algebraic trick I know, but I can't seem to find what they are actually doing.
I think I'm just missing some really simple algebra logic, but maybe I'm not understanding the steps to get Bézout coefficients?

Answer & Explanation

Bianca Chung

Bianca Chung

Expert

2022-07-18Added 16 answers

Step 1
Using the distributive property, which says that for any a,b,c,
a ( b + c ) = a b + a c ,
we can see that
3 1 ( 23 7 3 ) = 3 + ( 1 ) ( 23 + ( 7 ) ( 3 ) ) =
3 + ( 1 ) ( 23 ) + ( 1 ) ( 7 ) ( 3 ) = ( 1 ) ( 23 ) + 3 + ( 7 ) ( 3 ) = 1 23 + 8 3
Step 2
In our case, we had a = 1, b = 23, and c = 21 = ( 7 ) ( 3 )

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