A positive integer has k\:trailing\:zeros if its last k digits are all zero and it has a non-zero digit immediately to the left of these k zeros. For example, the number 1\:030\:000 has 4 trailing zeros. Define Z(m) to be the number of trailing zeros of the positive integer m. Lloyd is bored one day, so makes a list of the value of n-Z(n!) for each integer n from 100 to 10\:000, inclusive. How many integers appear in his list at least three times?
(Note: If n is a positive integer, the symbol n! (read “n factorial”) is used to represent the product of the integers from 1 to n. That is, n!= n(n-1)(n-2)\cdots(3)(2)(1). For example, 5!=5(4)(3)(2)(1) or 5!=120.)
Answer Choices
A. 2
B. 3
C. 4
D. 5
E. 6