$$\text{Prove true or false:}\quad M \bigotimes_{\mathbb{Z}} \mathbb{Z}_{21} \cong (M/3M) \bigoplus (M/7M)$$
$M$ is an abelian group, so a $\mathbb{Z}$ module I tried creating homomorphisms with the universal property of the tensor product for bilinear maps and prove that they are inverse to one another but im not getting the desire isomorphism, yet i dont know if the statement is true.
I've tried creating maps , for example one that sends $(m+3M,m_1 + 7 M)$ to $(m m_1 \bigotimes 1 + 21 \mathbb{Z})$ and one that sends $(m \bigotimes x+21\mathbb{Z})$ to $(xm + 3M, 1 +7M)$.
See that $\mathbb{Z}_{21}=\mathbb{Z}_3\oplus \mathbb{Z}_7$.
As tensor product distributes with direct sum, $$M\otimes_{\mathbb{Z}}\mathbb{Z}_{21}=M\otimes_{\mathbb{Z}}(\mathbb{Z}_3\oplus \mathbb{Z}_7)=(M\otimes_\mathbb{Z}\mathbb{Z}_3)\oplus (M\otimes_\mathbb{Z}\mathbb{Z}_7)$$
Can you take it from here?