On a criterion for the equality of Dedekind Sums

In [3] it was shown that the Dedekond sums $s(m_1,n)$ and $s(m_2,n)$ are equal only if $(m_1m_2-1)(m_1-m_2)\equiv 0$ mod $n$. Here we show that the latter condition is equivalent to $12s(m_1,n)-12s(m_2,n)\in \Z$. In addition, we determine, for a given number $m_1$, the number of integers $m_2$ in the range $0\le m_2<n$, $(m_1,m_2)=1$, such that $12s(m_1,n)-12s(m_2,n)\in \Z$, provided that $n$ is square-free.