  It is well known that for all $n\geq1$ the number $n+ 1$ is a divisor of the
central binomial coefficient ${2n\choose n}$. Since the $n$th central binomial
coefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unit
steps north or east, a natural question is whether there is a way to partition
these paths into sets of $n+ 1$ paths or $n+1$ equinumerous sets of paths. The
Chung-Feller theorem gives an elegant answer to this question. We pose and
deliver an answer to the analogous question for $2n-1$, another divisor of
${2n\choose n}$. We then show our main result follows from a more general
observation regarding binomial coefficients ${n\choose k}$ with $n$ and $k$
relatively prime. A discussion of the case where $n$ and $k$ are not relatively
prime is also given, highlighting the limitations of our methods. Finally, we
come full circle and give a novel interpretation of the Catalan numbers.
