Edit 2: Actually the wording of this is a bit off: "Thus ~ is not an equivalence relation because a~a does not necessarily imply gcd(a,a) > 1." Just say that 1~1 is not true therefore it's not an equivalence relation. Alternatively, say a~a is not true for all a in Z. Alternatively, say that gcd(a,a) > 1 is not true for all a in Z. Note that 1~1 not being true constitutes a counterexample.
----- Original response -----
Since it is not an equivalence relation, we give a counterexample.
gcd(2,6) = 2
gcd(6,15) = 3
gcd(2,15) = 1