The greatest common divisor of $a, b \in \mathbb{Z}$ can always be expressed as a linear combination of $a$, $b$. Furthormore, $\gcd(a,b)$ has the smallest magnitude of any number of the form $z = ax + by$
Please excuse my difficulties with mathematics. But this is something I keep coming back to again and again, and yet still can't reconcile this fact.
I understand the proof perfectly well... but if someone were to ask whats really going on here? or to give a heuristic/intuition on why this is really true, I wouldn't be able to answer.
In math, typically one has an idea, then uses a proof as a means of explaining why the idea is true. What I'm looking for, is the intuition behind the idea.
Thanks for any help in this, albeit embarrassingly simple problem.