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Let $p$ be an odd prime and $c$ a fixed integer with $(c,p)=1$. For each integer $a$ with $1\le a\le p-1$, it is clear that there exists one and only one $b$ with $0\le b\le p-1$ such that $ab\equiv c$ (mod $p$). Let $N(c,p)$ denote the number of all solutions of the congruence equation $ab\equiv c$ (mod $p$) for $1\le a$, $b\le p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1(modp)$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value problem involving...