Bézout's identity — Let $a$ and $b$ be integers with greatest common divisor $d$. Then there exist integers $x$ and $y$ such that $ax + by = d$. More generally, the integers of the form $ax + by$ are exactly the multiples of $d$.
Proof from Wikipedia: https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity
The proof is simple enough, but I was just wondering if anyone had anything more to say about why this theorem is to be expected. Why is a linear combination of two natural numbers a multiple of their greatest common divisor? Can this be understood in terms of the prime factorizations of the two natural numbers? Thank you.