It is now called Euclid's Algorithm. First, I will describe it using an example. We will find the gcd of 36 and 15. In this case, y is the gcd of a and b. Euclid's algorithm comes in handy with computers, because listing divisors is more difficult than the above algorithm.
Large numbers are difficult to factor, while they are relatively easy to divide.
Euclid did his proof of his algorithm geometrically, believe it or not, as algebra had not been invented yet. His algorithm is considered to be one of the best examples of an efficient algorithm. We divide top and bottom by their gcd. This is often handy to know in various branches of mathematics. But it is easier to use the lowest common denominator.