Consider the finite Abelian group $G = \mathbb Z_2 \times \mathbb Z_{12} \times \mathbb Z_{15}$. Find an isomorphic group a $G$ which is the direct product of only two cyclic groups. From this it can be deduced that $G$ can be generated by two elements. Give an example of such elements.
I suppose the order of $G$ is $360$, but I don't find two cyclic groups whose direct product makes $360$, I find: $360= 2^3 \times 5 \times 3^2$, but it's: $\mathbb Z _{24} \times \mathbb Z _{15}$, but only $15$ is cyclic...
Since $\gcd(2,15)=1$, we have, by the Chinese Remainder Theorem, that $$\Bbb Z_2\times\Bbb Z_{15}\cong\Bbb Z_{30},$$ which is cyclic; hence a group isomorphic to $G$ would be $$\Bbb Z_{12}\times \Bbb Z_{30}.$$