Take $k$ consecutive composite integers from a prime gap. What is known about the largest prime divisor of their product?

It seems to me that except for the triplet $(8,9,10)$, this largest prime divisor is always larger than $2k$, but I could not find an elementary (my level) proof. 

As $k$ grows it seems that the sharpness of $2k$ as lower bound is lost, therefore I expect that there should exist an "easy" argument. 

The question is essentially the same as [this one][1] from MSE but it did not get any answer. I could only find messy $k$-specific arguments for $k$ up to $4$ which I hope are correct.    


  [1]: http://math.stackexchange.com/questions/1495443/2k-as-a-lower-bound-for-the-largest-prime-divisor-of-binomial-coefficient-bi