Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$.
Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct:
(i) First we want a projective resolution of $M$: $ 0 \to \mathbb Z \xrightarrow{\cdot 284 } \mathbb Z \xrightarrow{\pi} M \to 0$ Then we chop off $M$ to get $ 0 \to \mathbb Z \xrightarrow{d_1 = \cdot 284 } \mathbb Z \xrightarrow{d_0 = 0} 0$ And apply $- \otimes N$ to get $ 0 \to \mathbb Z \otimes N \xrightarrow{d_1^\ast } \mathbb Z \otimes N \xrightarrow{d_0^\ast} 0$
(ii) Now we see that $\operatorname{Tor_i}{(M,N)} = 0$ for $i \geq 2$.
(iii) We know that $\operatorname{Tor_0}{(M,N)} = M \otimes N = \mathbb Z / 284 \mathbb Z \otimes \mathbb Z / 2 \mathbb Z$
(iv) $\operatorname{Tor_1}{(M,N)} = \operatorname{Ker}{d_0^\ast} / \operatorname{Im}{d_1^\ast} = 0 / 0 = 0$.
Thanks for your help.