2.1.2 Euclidean Algorithm Video
PROFESSOR: The greatest common divisor of two numbers is easy to compute. And that's a factor will play a crucial role in the number theory we're going to develop, and the properties of some of the modern codes that are based on number theory. The efficient way to compute the GCD of two numbers is based on a classical algorithm known as Euclidean algorithm, which is several thousand years old. And let's describe how it works now. So the Euclidean algorithm is based on the following lemma, which we'll call the remainder lemma, and it says that if a and b are two integers, then the greatest common divisor of a and b is the same as the greatest common divisor of b, and the remainder of a divided by b-- providing, of course, b is not 0, because otherwise you can't divide by b. OK how do you make sense out of this? Why is this true? Well, it's actually a very easy proof. Remember that by the so-called division algorithm-- or it's really a theorem-- ...