Finding an isomorphism from $\Bbb Z_{540}$ to a direct product of cyclic groups

Question

$\Bbb Z_{540} \cong \Bbb Z_{2^2} \times \Bbb Z_{3^3} \times \Bbb Z_{5} $
$\varphi : \Bbb Z_{540} \to \Bbb Z_{2^2} \times \Bbb Z_{3^3} \times \Bbb Z_{5}$
$\varphi ([k]_{540}) = ([k]_{2^2},[k]_{3^3},[k]_{5})$ is iso .
I think this is true from the Chinese remainder theorem but I'm not certain

Answer 1

Yes, if you write $[k]_n=k ~(~mod~n)$ this is indeed an isomorphism by the CRT.