Equality of Dedekind sums modulo 24$\mathbb{Z}$

Let $S(a,b)=12s(a,b)$, where $s(a,b)$ denotes the classical Dedekind sum. In a recent note E. Tsukerman gave a necessary and sufficient condition for $S(a_1,b)-S(a_2,b)\in 8\mathbb{Z}$. In the present paper we show that this condition is equivalent to $S(a_1,b)-S(a_2,b)\in 24\mathbb{Z}$, provided that $9\nmid b$. Tsukerman also obtained a congruence mod 8 for $bT(a,b)$, where $T(a,b)$ is the alternating sum of the partial quotients of the continued fraction expansion of $a\mathrm{/}b$. We show that the respective congruence holds mod $24$ if $3\nmid b$ and mod $72$ if $3\mid b$.