29th Eötvös 1925 Problem 2 How many zeros does the the decimal representation of 1000! end with?
There are 200 multiples of 5 in X = {1, 2, ... , 1000}. 40 of them are multiples of 5 2 , so each bring an additional power of 5. 8 are multiples of 5 3 , so each bring another, and 1 is a multiple of 5 4 , so bringing another. Thus the highest power of 5 dividing 1000! is 5 249 . Obviously 1000! is divisible by 2 249 , so the highest power of 10 divising 1000! is 10 249 . So it ends in 249 zeros. 29th Eötvös 1925 © John Scholes jscholes@kalva.demon.co.uk 1 Nov 2003 Last corrected/updated 1 Nov 03