Open course ware

PROFESSOR: Let's continue our examination of GCD's and linear combinations and Euclidean algorithm by examining what's often called the extended Euclidean algorithm. It's a good name for it. Its ancient name, dating back to ancient India, is the pulverizer. And we will see what that does in a moment. So the theorem that is the culmination that we're aiming for is that the GCD of two numbers is an integer linear combination of the two numbers. That is, the GCD of a and b is simply sa plus tb, where s and t are integers. And what the pulverizer enables us to do is given a and b we can find s and t. In fact, we can find s and t virtually as efficiently as the Euclidean algorithm. It's just by performing the Euclidean algorithm and keeping a track of some additional side information as it progresses. Now a corollary of this fact is that we now know that the-- if we want to characterize the linear combinations of a and b, they're precisely the multi...