discrete mathematics, Euclidean algorithm

Question

I'm having troubles to find the coefficients: 1357 + 49 = 1 = (1357,49). I started solving the equations by the Euclidean algorithm, but I can't replace them all. Can someone help me?

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From then on I couldn't do it anymore.

Welcome to Mathematics Stack Exchange. You should show your attempt — J. W. Tanner, Jun 13 '23 at 19:11
Show at least the beginning of the Euclidean Algorithm you did, and I will complete that...The resulting numbers are not that low. — DonAntonio, Jun 13 '23 at 19:18
You should write the numbers in the post to avoid down-votes and votes to close. A cursory glance at the work in your link shows that this is a perfectly acceptable question but that can't be seen from what you actually posted. — John Douma, Jun 13 '23 at 19:40
On the second-to-last equation, did you mean $1=4\cdot34-\color{red}9\cdot15$? — J. W. Tanner, Jun 13 '23 at 19:41
As @J.W.Tanner points out, you just made an arithmetic mistake on the penultimate line. — John Douma, Jun 13 '23 at 19:42
You can also verify yourself that equality no longer holds since which line. — peterwhy, Jun 13 '23 at 19:48
Welcome to MathSE. This MathJax tutorial explains how to typeset mathematics on this site. — N. F. Taussig, Jun 14 '23 at 08:39
Duplicate of Extended Euclidean Algorithm: backward and forward form — Bill Dubuque, Sep 20 '23 at 21:06

score 1 · Answer 1 · answered Jun 13 '23 at 20:45

1

$$\begin{cases} 1357=49\cdot27+34\\{}\\ 49=34\cdot1+15\\{}\\ 34=15\cdot2+4\\{}\\ 15=4\cdot3+3\\{}\\ 4=3\cdot1+1\\{}\\ 3=1\cdot3+0\;\text{(check case)}\end{cases}$$

And now read from bottom to top:

$$1=4-3=4-(15-4\cdot3)=4\cdot4-15=4(34-15\cdot2)-15=4\cdot34-9\cdot15=$$

$$=4\cdot34-9(49-34)=13\cdot34-9\cdot49=13(1357-49\cdot27)-9\cdot49$$

$$=13\cdot1357-360\cdot49$$

answered Jun 13 '23 at 20:45

DonAntonio

211,718
17
136
287

Thank you so much! – Mariana Monteiro Jun 14 '23 at 00:38

discrete mathematics, Euclidean algorithm

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