I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/65520$.

This looks like a good example to stop and ask whether there is any explanation for this $K$-group of integers (23 is just an arbitrary fixed number for this purpose). By "explanation" I mean a reasoning that would allow to find at least some properties of $K_{23}$ in advance of computing it or some intuition behind the result.

Here's one thing I already know:

• non-torsion part of $K(\mathbb Z)$ is $\mathbb Z[0,5,9,13...]$ so $K_{23}(\mathbb Z)$ is pure torsion