The Euclidean algorithm

In this section, we will develope a method for computing $gcd(a,b)$. This of practical importance.

Another motivation for this section arises from the following fact. We have seen that all the multiples of a single integer $b$ may be visualized in several ways: as a series of ``bumps'' spaced $b$ units apart on the real number line, or as the first column of the array of integers when arranged in a series of increasing rows of $b$ numbers each (see (1.1)). The following question asks: How do we generalze this?

Question: Let $a$ and $b$ be any two non-zero integers. Can the set of all possible sums of multiples of $a$ and $b$ be described in a ``simple'' way? If so, what is it?