Let d = gcd(a,b). Then d is a common divisor of a and b. By the extended Euclidean algorithm d = xa + yb for some integers x and y. If c is any common divisor of a and b, then it also divides d. This proves that the first assertion implies the second.

As for the other way around, suppose that d is as in the second statement. Since gcd(a,b) is a common divisor of a and b it must divide d. On the other hand d cannot be larger than the greatest common divisor of a and b. Hence d and gcd(a,b) must be equal. This proves that the second statement implies the first.

Let d = gcd(a,b) and let e be the least positive integer that can be expressed as xa + yb with integers x and y. We show that d = e. Since d is a common divisor of a and b the equality e = xa + yb implies that d divides e. So d e. Moreover, d itself can also be written as a linear combination of a and b. So d e by the defining property of e. Hence d must be equal to e. This proves the equivalence.