Converting Linear Combination to GCD

Question

How would I convert a linear combination to a GCD? I was told to use the Euclidean Algorithm and I know how to convert from GCD to linear combination, but not the reverse.

Edit: linear combination as in $ax+by=c$ given $a$, $b$, and $c$ and assuming all values are integer values

Answer 1

Hint $\,e:=c/d\,$ times $\,\gcd(a,b)=d \Rightarrow \bbox[5px,border:1px solid #c00]{\gcd(ae,be) = c}\,\ $ Note $\,d\mid a,b\,\Rightarrow\,d\mid c=ax\!+\!by$