Page "Euclidean algorithm" Paragraph 7

Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers until one of them is zero.

When that occurs, the GCD is the remaining nonzero number.

By reversing the steps in the Euclidean algorithm, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e. g., 21 = × 105 + × 252.

This important property is known as Bézout's identity.