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

This looks like a good point 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 this group 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 $\mathbb Z$ in degrees $0,5,9,13,\dots\ $ so $K_{23}(\mathbb Z)$ is pure torsion

Wikipedia says that "The torsion subgroups of $K_{2i+1}(\mathbb Z)$ ... have recently been determined."

Update: I learned from the article by Soule how this number $=2 * 12 * 2730 $ where 2730 is the denominator of 12-th Bernoulli number. But the question stands.