Abstract Algebra: External Direct Products?

Question

How do you show if the external direct product of Z_3 and Z_9 is isomorphic to Z_27?

Answer 1

The first response to this question is correct.
If there was an isomorphism from Z3XZ9 to Z27 then let
the isomorp send (a,b) to 1 in Z27. But a in Z3 and b in Z9 means that 9a (modular product but 9 a's)= 3*(3a) = 0 and 9b = 0 so 9(a,b) = 0 but, the sum of 9 1's in Z27 is 9 which is not 0. So can't be an isomorph.

Answer 2

It can't possibly be. Z_27 contains an element of order 27 (namely, 1), whereas no element in Z_3×Z_9 has order greater than 9.