Working Backwards from the GCD: Finding the Solution to a GCD Calculation

[Math100]

Homework Statement

Find a solution of 621m+483n=k, where k is the gcd of 621 and 483.

Relevant Equations

None.

First, we start to calculate the gcd(621, 483).
Applying the Euclidean algorithm produces:
621=1*483+138
483=3*138+69
138=2*69.
Thus gcd(621, 483)=69.

And now I'm stuck, because I don't know how to find the solution of this after finding out the gcd of 621 and 483. I was told to work/calculate backwards starting from 69, for example, like this:

69=483-(3*138)
?=?

 

[fresh_42]

Start at the end:
\begin{align*}
138=2\cdot 69 & \Longrightarrow 483=3\cdot 138+69 =3\cdot(2\cdot 69)+69=7\cdot 69 \\
483=7\cdot 69 & \Longrightarrow 621=1\cdot 483 +138 = \ldots
\end{align*}

 

[Math100]

Start at the end:
$$
138=2\cdot 69 \Longrightarrow 483=3\cdot 138+69 =3\cdot(2\cdot 69)+69=7\cdot 69
483=7\cdot 69 \Longrightarrow 621=1\cdot 483 +138 = \ldots
$$

Sorry, I don't understand this, as this was written in latex.

 

[fresh_42]

Sorry, I don't understand this, as this was written in latex.

Now it renders. Refresh the page (F5).

 

[fresh_42]

At the end, you can use $4\cdot 7 - 3\cdot 9 =1$ and multiply it with $69.$

 

[Math100]

Thank you for the help.