1992 AIME Problems/Problem 15
Contents
Problem
Define a positive integer to be a factorial tail if there is some positive integer such that the decimal representation of ends with exactly zeroes. How many positive integers less than are not factorial tails?
Solution
Let the number of zeros at the end of be . We have .
Note that if is a multiple of , .
Since , a value of such that is greater than . Testing values greater than this yields .
There are distinct positive integers, , less than . Thus, there are positive integers less than that are not factorial tails.
Solution 2
After testing various values of in of solution 1 to determine for which , we find that . WLOG, we select . Furthermore, note that every time reaches a multiple of , will gain two or more additional factors of and will thus skip one or more numbers.
With this logic, we realize that the desired quantity is simply , where the first term accounts for every time number is skipped, the second term accounts for each time numbers are skipped, and so on. Evaluating this gives us . - Spacesam(edited by srisainandan6)
See also
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