Divisible by this year? (Part 3: What if I did this?)

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 $2014!! = 1^{2014} \times 2^{2013} \times \ldots \times 2014$ .

We begin by checking all factors of $2014!!$ of the form $(53n)^{2015-53n}$ ; they have $53^{2015-53n}$ as a factor, and there are 38 such integers; hence, a lower bound for $k = \displaystyle \sum_{n=1}^{38} 2015-53n = 37297$ .

Then, we check all factors of the form $(53^{2}n)^{2015-53^{2}n}$ ; however, because $53^{2} > 2014!$ , there are no more factors to check and hence our answer is $k = \boxed{37297}$ .