Eötvös 1925/2 solution

29th Eötvös 1925

Problem 2

How many zeros does the the decimal representation of 1000! end with?

Solution

There are 200 multiples of 5 in X = {1, 2, ... , 1000}. 40 of them are multiples of 5², so each bring an additional power of 5. 8 are multiples of 5³, so each bring another, and 1 is a multiple of 5⁴, so bringing another. Thus the highest power of 5 dividing 1000! is 5²⁴⁹. Obviously 1000! is divisible by 2²⁴⁹, so the highest power of 10 divising 1000! is 10²⁴⁹. So it ends in 249 zeros.

29th Eötvös 1925