Show that $2n\choose n$ is divisible by $2n-1$ I have found many questions asking to prove $2n\choose n$ is divisible $2$, but I also observed by trying the first few "$n$" that $2n\choose n$ is divisible  by $2n-1$. 
It sure seems true when $2n-1$ is prime, but is it true in general for all $n$?

Simply expand  and seperate the factorial in the numerator. 

Clearly,  is a factor in the numerator. Therefore,  is indeed divisible by .
