My generalization to a problem , not sure it is true

I am doing the past Putnam problems and one is

Prove that $\displaystyle \frac{1}{n+1} \binom{2n}{n} $ is an integer .

I make a generalization , which is :

$\displaystyle \frac{1}{n+k} \binom{2n}{n} $ is an integer iff $\displaystyle 0 < k < \frac{x}{2} $ , where $\displaystyle x $ is the minimun prime factor of $\displaystyle n+k $ .

If it is true , then the problem from Putnam is just a special case . However , I couldn't trust my proof so i beg for help here .

Thanks

p.s. I make use of floor function to prove it .

EDIT: I find some errors in my statement , iff should be changed to if and the equality can hold . i.e. $\displaystyle 0<k \leq\frac{x}{2} $