Congruences mod 4 for the alternating sum of the partial quotients

We apply a technique used in $[$Tsukerman, Equality of Dedekind sums mod $\mathbb{Z}$, $2\mathbb{Z}$ and $4\mathbb{Z}$, arXiv:1408.3225] combined with the Barkan-Hickerson-Knuth-formula in order to obtain congruences mod $4$ for the alternating sum of the partial quotients of the continued fraction expansion of $a\mathrm{/}b$, where $0<a<b$ are integers.