Divisible by this year??? (Part 4: INSANITY!!!!!!!!)

In this question, we only need to consider the prime factors of 2014: 2, 19, and 53.

Since 2, 19, and 53 are pairwise coprime , we will not overcount and thus we can further simplify the question by looking at the maximal $k$ for which $53^k | 2014!!!$ .

Notice that $2014!!! = 1^{2014(2015)/2} \times 2^{2013(2014)/2} \times \ldots \times 2014$ since $\displaystyle \sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ .

We begin by checking all factors of $2014!!!$ of the form $(53n)^{(2015-53n)(2016-53n)/2}$ ; they have $53^{(2015-53n)(2016-53n)/2}$ as a factor, and there are 38 such integers; hence, a lower bound for $k = \displaystyle \sum_{n=1}^{38} \frac{(2015-53n)(2016-53n)}{2} = 24740014$

Then, we check all factors of the form $(53^{2}n)^{(2015-53^{2}n)(2016-53^{2}n)/2}$ ; however, because $53^{2} > 2014$ , there are no more factors to check and hence our answer is $k = \boxed{24740014}$ .