Long exact sequence for Ext

A short exact sequence

be a short exact sequence of left $R$-modules, and let $A$ be a left $R$-module. Then there is a natural long exact sequence

1. Computing $\operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/n, A)\cong A/nA$. Start from
   $$0\to \mathbb Z \xrightarrow{\cdot n}\mathbb Z \to \mathbb Z/n \to 0.$$
   Apply $\operatorname{Hom}_{\mathbb Z}(-,A)$. Since $\operatorname{Hom}_{\mathbb Z}(\mathbb Z,A)\cong A$, the relevant piece of the long exact sequence is
   $$A \xrightarrow{\cdot n} A \to \operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/n,A)\to 0,$$
   so
   $$\operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/n,A)\cong \operatorname{coker}(\cdot n:A\to A)\cong A/nA.$$
   (Also $\operatorname{Ext}^i_{\mathbb Z}(\mathbb Z/n,-)=0$ for $i\ge 2$ because $\mathbb Z/n$ has a length-1 projective resolution.)

1. Computing $\operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/n, \mathbb Z/m)\cong \mathbb Z/\gcd(n,m)$. Take $A=\mathbb Z/m$ in the previous example:
   $$\operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/n,\mathbb Z/m)\cong (\mathbb Z/m)/n(\mathbb Z/m) \cong \mathbb Z/\gcd(n,m).$$

These sequences are instances of the long exact sequence for derived functors, with connecting maps supplied by the connecting homomorphism.