The extended Euclidean algorithm leads us to the following characterization of the gcd.
Let a and b be two positive integers.
Characterization of the gcd
The following three statements are equivalent.
- gcd(a,b) = d.
- d is a positive common divisor of a and b such
that any common divisor c of a and b is a divisor of d.
- d is the least positive integer that can be
expressed as xa + yb with integers x and y.