  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
$a/b$, where $0<a<b$ are integers.
