Problem: Determine the largest positive integer such that 1005 ! is divisible by .
Answer Choices:
A.
B.
C.
D.
E. none of these answers
Solution:
If is the factorization of 1005 ! into powers of distinct primes, then is the minimum of and . Now 201 of the integers between 5 and 1005 are divisible by 5 ; forty of these 201 integers are divisible by ; eight of these forty integers are divisible by ; and one of these eight integers is divisible by . Since .