2006 AIME I Problems/Problem 4
Let be the number of consecutive 's at the right end of the decimal representation of the product Find the remainder when is divided by .
A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and the number of 2s dividing into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.
One way to do this is as follows: of the numbers have a factor of . have a factor of . have a factor of . And so on. This gives us an initial count of . Summing this arithmetic series of terms, we get . However, we have neglected some powers of - every term for has an additional power of dividing it, for extra; every n! for has one more in addition to that, for a total of extra; and similarly there are extra from those larger than and extra from . Thus, our final total is , and the answer is .
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