I find myself able to mechanically apply the "extended" Euclidean algorithm to find the gcd of two integers and to write a linear combination by working backwards. However, I do not have a good grasp as to why this works. Artin gives a paragraph-long explanation of it, but I am not able to follow it, nor do I have any good intuition for why this process works.
One can compute a greatest common divisor easily by repeated division with remainder. For example, if $a = 314$ and $b = 136$, then $$314 = 2 \cdot 136 + 42, \; 136 = 3 \cdot 42 + 10, \; 42 = 4 \cdot 10 + 2.$$ Using the first of these equations, one can show that any integer combination of $314$ and $136$ can also be written as an integer combination of $136$ and the remainder $42$, and vice versa. So $\mathbb{Z}(314) + \mathbb{Z}(136) = \mathbb{Z}(136) + \mathbb{Z}(42)$, and therefore $\gcd(314, 136) = \gcd(136, 42)$. Similarly, $\gcd(136, 52) = \gcd(42,10) = \gcd(10,2) = 2$. So the greatest common divisor of $314$ and $136$ is $2$. This iterative method of finding the greatest common divisor of two integers is called the Euclidean Algorithm.
I'm fine with the first line. If I have an integer combination $$314x + 136y,$$ I can use the first given equation to write instead $$314x + 136y = (2 \cdot 136 + 42)x + 136y = 2x \cdot 136 + 42x + 136y = (2x+y) \cdot 136 + 42x.$$ I can then write $136$ in terms of $42$ and $10$ and get a linear combination of $42$ and $10$, and so forth, and since the integers are closed, my coefficients will always be integers, so I end up with another integer combination.
I cannot understand anything beyond this point in the text, though. I do not know what Artin means by the notation $\mathbb{Z}(314)$. Further, I have no intuition for why the $\gcd$ remains constant all the way down the chain or, even, why the last remaining, non-zero remainder in the algorithm is the $\gcd$.
Any help with the intuition would be greatly appreciated.