That's a weird question: a "greatest" element of anything is always unique. It should seem obvious to you when you think of the set of the common divisors: this is a finite set of integers, it must have a greatest element, and I can't see how one may wonder if it is unique. Here's a proof anyway.

Suppose is a greatest element of a nonempty set (here, is the set of common divisors of two non-zero integers, it is nonempty since it contains 1). Then if is another greatest element, we would have both (because is larger than any element of ) and (because is larger than any element of ), hence .