Q:

What am I missing here: $U(144) \neq U(140)$

I'm confused about the following exercise:
Prove that $U(144)$ is isomorphic to $U(140)$.
Here are my thoughts:
$$U(144) = U(12^2) = U(3^2)\oplus U(2^4) = \mathbb Z_{6} \oplus \mathbb Z_{8}$$
and
$$ U(140) = U(2^2 *7*5) = U(2^2) \oplus U(5) \oplus U(7) = \mathbb Z_{2} \oplus \mathbb Z_{4} \oplus \mathbb Z_{6}$$
And I have the following result:
$\mathbb Z_{n_1 \dots n_k}\cong \mathbb Z_{n_1}\oplus \dots \oplus \mathbb Z_{n_k}$ if and only if $n_i$ are pairwise coprime.
But $2$ and $4$ are not coprime therefore $\mathbb Z_{2} \oplus \mathbb Z_{4} \oplus \mathbb Z_{6}$ is not isomorphic to $\mathbb Z_{6} \oplus \mathbb Z_{8}$. 

What am I doing wrong?

A:

Hint: $U(2^4)$ is not isomorphic to $\mathbb{Z}_8$. 

