Question

if 1000 is a factor of n, what other positive integers divide n? how many such integers are there?

Answer 1

N = 1000! = 1000 × 999 × 998 × 997 × · · · × 3 × 2 × 1. N is divisible by 5,25,125,625,···. Each of these factors is a power of 5. That is,
5 = 51, 25 = 52, 125 = 53, 625 = 54, and so on. Determine the largest power of 5 that divides N.

Answer 2

In order to determine the largest power of 5 that divides N, we need to count the number of times the factor 5 appears in the factorization of N.
First, let’s look at the numbers that are divisible by 5 in N!. Each of the
numbers {5, 10, 15, 20, . . . , 990, 995, 1000} contains a factor of 5. That is a total
of 1000 = 200 factors of 5. 5
Those numbers that are multiples of 25 will add an additional factor of 5, since 25 = 5 × 5. There are 1000 = 40 numbers less than or equal to 1000 which are
25
multiples of 25. So we gain another 40 factors of 5 bringing the total to 200 + 40 = 240.
Those numbers that are multiples of 125 will add an additional factor of 5.
This is because 125 = 5 × 5 × 5 and two of the factors have already been
counted when we looked at 5 and 25. There are 1000 = 8 numbers less than or 125
equal to 1000 which are multiples of 125. So we gain another 8 factors of 5 bringing the total to 240 + 8 = 248.
Those numbers that are multiples of 625 will add an additional factor of 5. This is because 625 = 5 × 5 × 5 × 5 and three of the factors have already been counted when we looked at 5, 25 and 125. There is 1 number less than 1000 which is a multiple of 625 (namely, 625). So we gain another factor of 5 bringing the total to 248 + 1 = 249. So, when N is factored there are 249 factors of 5. Therefore, the largest power of 5 that divides N is 5^249.
the answer is 5^249.