  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 $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and
$K_{2k}(\mathbb{Z}[\zeta_m])$, where $\zeta_m$ is a primitive $m$th root of
unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(\zeta_l)$ 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 $p$. This generalizes a
previous result of Browkin which in turn has applications towards establishing
the existence of infinitely many cyclic extensions of degree $p$ which are $(p,
p-1)$-rational.
