13. If $2010!=M \times 10^{k}$, where $M$ is an integer not divisible by 10 , find the value of $k$.
13. Answer: 501
The number $k$ is the number of the factor 10 that occurs in 2010 !. This number is given by the number of pairs of prime factors 2 and 5 in 2010!.
Now between 1 and 2010 , there are:
- 402 integers with 5 as a factor;
- 80 integers with 25 as a factor;
- 16 integers with 125 as a factor;
- 3 integer's with 625 as a factor.

Therefore the total number of prime facţor 5 in 2010 ! is $402+80+16+3=501$. As there are clearly more than 501 prime factor 2 in 2010 !, we obtain $k=501$.