  A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for
integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive
divisors of $n$. In this paper, we establish the structure theorem of odd
multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We
prove the divisibility of the Euler part of odd multiperfect numbers and
characterize the forms of odd perfect numbers $n=\pi^\alpha M^2$ such that
$\pi\equiv\alpha(\text{mod}8)$. We also present some examples to show the
nonexistence of odd perfect numbers as applications.
