Question 494976
2011!!! is equal to 2011*2008*2005*...*4*1. The factors that are multiples of 5 are {2005, 1990, 1975, ..., 10}, and the factors that are multiples of 2 are {2008, 2002, ..., 4}. Hence, we can compute the largest power of 5 that divides 2005*1990*1975*...*10 (there will be sufficient 2's).


There are 134 elements in the set {10, 25, ..., 2005}. However, we need to account for multiples of 25, 125, and 625. The multiples of 25 in that set are {25, 100, 175, ..., 1975} (simply all multiples of 25 that are 1 mod 3; 27 numbers); the multiples of 125 are {250, 625, ..., 1750} (5 numbers), and the multiples of 625 are {625} (1 number). It suffices to simply add the sizes of each set, so that we count multiples of 25 twice, multiples of 125 three times, and multiples of 625 four times. Hence, 134+27+5+1 = 167, so k = 167.