Find the largest integer $n$ such that $10^n$ divides $10^6!$

Question

Let $N=10^6!$

1. Find the largest integer $n$ such that $10^n$ divides $N$.

2. Furthermore, compute the first digit and the last non-zero digit of $N$.

I have some ideas that you should be able to use the factors 2 and 5 of 10 and modular arithmetic to solve it but I'm not really getting anywhere.

Thanks!

Answer 1

Hint: you need to count how many powers of 10 you can form out of the prime factorization of N. You will have many more twos than fives (why?), so you just need to determine the exponent of 5 in the prime factorization of N.

To determine the exponent of 5 in the prime factorization, think about where the fives are going to come from: namely, multiples of 5. You'll have to be careful, though. Some multiples of 5 will contribute more fives than others...