GCD as linear combination of two numbers

I know you can write gcd of two numbers as a linear combination of two numbers, but my question is what do we achieve by doing that? is there any significance of writing gcd as linear combination? does that give us some more interesting info about gcd and those two numbers?

• In many arithmetic algorithms, it's handy to be able to invert a number $a$ modulo another number $n$, that is to solve $ax\equiv1\pmod n$. Jan 20, 2018 at 11:43

• @bhavindhedhi for example $gdc(4,3)=1$ and $-2*4+3*3=1$ but also $4*4-5*3=1$