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It is well known that for all $n\geq1$ the number $n+ 1$ is a divisor of thecentral binomial coefficient ${2n\choose n}$. Since the $n$th central binomialcoefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unitsteps north or east, a natural question is whether there is a way to partitionthese paths into sets of $n+ 1$ paths or $n+1$ equinumerous sets of paths. TheChung-Feller theorem gives an elegant answer to this question. We pose anddeliver 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 generalobservation regarding binomial coefficients ${n\choose k}$ with $n$ and $k$relatively prime. A discussion of the case where $n$ and $k$ are not relativelyprime is also given, highlighting the limitations of our methods. Finally, wecome full circle and give a novel interpretation of the Catalan numbers.

On a divisor of the central binomial coefficient