Extended Euclidean Algorithm

The algorithm, which is used to find the greatest common divisor of two integers

The Euclidean algorithm is usually used simply to find the greatest common divisor of two integers. (For a description of this algorithm, see the notes about additional topics in number theory.) The standard Euclidean algorithm gives the greatest common divisor and nothing else. However, if we keep track of a bit more information as we go through the algorithm, we can discover how to write the greatest common divisor as an integer linear combination of the two original numbers. In other words, we can find integers s and t such that

gcd(a, b) = sa + tb.

[Note that, since gcd(a, b) is usually less than both a and b, one of s or t will usually be negative.]

As a reminder, here are the steps of the standard Euclidean algorithm to find the greatest common divisor of two positive integers a and b:

Set the value of the variable c to the larger of the two values a and b, and set d to the smaller of a and b.

Find the remainder when c is divided by d. Call this remainder r.

If r = 0, then gcd(a, b) = d. Stop.

Otherwise, use the current values of d and r as the new values of c and d, respectively, and go back to step 2.

The extended Euclidean algorithm uses the same framework, but there is a bit more bookkeeping. Before we present a formal description of the extended Euclidean algorithm, let’s work our way through an example to illustrate the main ideas.