Find the gcd and write it a linear combination

Question

$$24,-11$$

So we begin with $gcd(24,-11)$

$$24=-2*-11+2=gcd(-11,2)$$

$$-11=-6*2+1=gcd(2,1)=1$$

So $gcd(24,-11)=1$ now how do I find x,y such that $1=24x-11y$?

You are severely abusing the $=$ sign here. – Nick Peterson Dec 19 '16 at 20:41 — Nick Peterson, Dec 19 '16 at 20:41

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

2

Easier than back-substitution is too keep track of the linear relations from the start

$$\begin{align} 24 &= \ \ 1(24)+0(11)\\ 11 &=\ \ 0(24)+1(11)\\ 2 &=\ \ 1(24)-2(11)\ \ =\ \rm row_1 - 2\cdot row_2\\ 1 &=\! -5(24)\!+\!11(11)\, = \ \rm row_2 - 5\cdot row_3 \end{align}$$

i.e. we perform the Euclidean algorithm on the LHS and in parallel on the RHS.

See this answer for much more on this version of the extended Euclidean algorithm.

edited Apr 13 '17 at 12:21

Community

1

answered Dec 19 '16 at 20:48

Bill Dubuque

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Awesome! thanks a lot, I will use this! – gbox Dec 19 '16 at 20:57
The answer in the book is $1=624+13(-11)$ – gbox Dec 20 '16 at 06:44

score 1 · Answer 2 · answered Dec 19 '16 at 20:46

The standard trick is to substitute backward from the equations you already have. For example, you have $-11=-6\cdot2+1$, so that means $1=-11+6\cdot2$. Using your first equation, we can re-write $2$ to yield the new equation $1=-11+6\cdot(24+2\cdot-11)$, which implies $\boxed{1=6\cdot24-11\cdot13}$ by distributing the $6$ and combining like terms.

Find the gcd and write it a linear combination

2 Answers2