19 is prime.


2016 ÷ 19 = 106 R2
106 × 19 = 2014

We can get 106 times of 19 at most. This means there are 106 numbers which are multiples of 19 so they each contain at least one prime factor 19.

The first of them is number 19 itself and the 106th is the number 2014.

(Basically we have 19 x 1 = 19, 19 x 2 = 38, 19 x 3 = 57,... 19 x 105 = 1995 , 19 x 106 = 2014)


Now some of these numbers contain more than one prime factor of 19.

Since these 106 numbers are multiples of 19, we want those numbers which contain a second 19 as their factors, in order to include them as well.


106 ÷ 19 = 5 R11

So we have 5 such numbers.

(i.e 19 x 19 x 1 , 19 x 19 x 2, 19 x 19 x 3, 19 x 19 x 4, 19 x 19 x 5)


Total number of factors 19 we can get = 106 + 5 = 111


So the highest 19ⁿ that divides 2016! is 19¹¹¹