  In a previous it was shown that the Dedkind sums $12s(m,n)$ and $12s(x,n)$,
$1\le m,x\le n$, $(m,n)=(x,n)=1$, are equal mod $\Z$ if, and only if,
$(x-m)(xm-1)\equiv 0$ mod $n$. Here we determine the cardinality of numbers $x$
in the above range that satisfy this congruence for a given number $m$.
