@Koro Who would ask it unless the person is a beginner or a university student seeing Number Theory in its most beautiful form for the first time?

@Koro Btw I tried doing the sort of random testing in Bezout's identity with $a = 6, b =3$. As we know, $g = (a,b) = 3$; also we're interested in finding the least positive integer value of $6x + 3y$. Now suppose that there was an element smaller than the GCD that is positive. We can write the inequality $0 < 6x + 3y < 3 \implies 0 < 2x + y < 1$. We see, there can't be an integer between $0$ and $1$, hence the least possible value of the expression must be $3$.

I think that itself is a consequence of Bezout's identity with a pair of coprime integers (here, $2$ and $1$)

@Koro The question I mentioned is the one in the third edited comment in the comment block before your reply

So if you were to try and find the least positive integer value for $ax + by, (a,b) = 1$ and you had no idea about Bezout's identity, you might try to do it as an optimisation problem and yet obtain the same results.