Is Z8 Isomorphic to Z4 x Z2?

[Dahaka14]

Homework Statement

Show that $$\mathbb{Z}_{8}$$ is not isomorphic to $$\mathbb{Z}_{4}\times\mathbb{Z}_{2}$$.

Homework Equations

$$\mathbb{Z}_{mn}\cong\mathbb{Z}_{m}\times\mathbb{Z}_{n}\iff \gcd(m,n)=1$$

The Attempt at a Solution

I would say that since $$\gcd(4,2)\neq1$$, they are not isomorphic.

There must be something that I am misunderstanding though since $$\mathbb{Z}_{4}$$ is not isomorphic $$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$$...

 

[Dahaka14]

Wow...$$\gcd(2,2)=2$$...awful. Sorry for wasting space.

 

[Deveno]

shows that Z8 has an element of order 8, and Z4 x Z2 does not.