Any problem in P can be reduced to the language of odd integers
Let $S$ be any language in $\mathsf{P}$. You are looking for a function $f$ with the following properties:

$f$ can be computed in polynomial time.
If $x \in S$ then $f(x) \in A$.
If $x \notin S$ then $f(x) \notin A$.

Since $S$ is in $\mathsf{P}$, we can determine whether $x \in S$ in polynomial time. Therefore, the reduction $f$ can work as follows:

Determine whether $x \in S$.
If $x \in S$ then output something in $A$.
If $x \notin S$ then output something not in $A$.

You take it from here.