On the $p$-divisibility of even $K$-groups of the ring of integers of a cyclotomic field

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[ζ_m+ζ_m^{-1}])$ and $K_{2k}(\mathbb{Z}[ζ_m])$, where $ζ_m$ is a primitive $m$th root of unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(ζ_l)$ for some prime $l$, we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal{O}_F)$ to be divisible by $p$.