12. Let D D be a UFD. An element d ∈ D d∈D is a greatest common divisor of a a and b b in D D if d ∣ a d∣a and d ∣ b d∣b and d d is divisible by any other element dividing both a a and b . b. If D D is a PID and a a and b b are both nonzero elements of D , D, prove there exists a unique greatest common divisor of a a and b b up to associates. That is, if d d and d ′ d′ are both greatest common divisors of a a and b , b, then d d and d ′ d′ are associates. We write gcd ( a , b ) gcd(a, b) for the greatest common divisor of a a and b .