Difference between revisions of "2006 AIME I Problems/Problem 4"
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== See also == | == See also == | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
| + | * [[Combinatorics]] | ||
| + | * [[Number Theory]] | ||
[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 16:17, 18 July 2006
Problem
Let 
be the number of consecutive 0's at the right end of the decimal representation of the product 
Find the remainder when is divided by 1000.
Solution
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: 96 of the numbers 1!, 2!, 3!, ..., 100! have a factor of 5. 91 have a factor of 10. 86 have a factor of 15. And so on. This gives us an initial count of 
. Summing this arithmetic series of 20 terms, we get 970. However, we have neglected some powers of 5 -- every n! term for 
has an additional power of 5 dividing it, for 76 extra; every n! for 
has one more in addition to that, for a total of 51 extra; and similarly there are 26 extra from those larger than 75 and 1 extra from 100. Thus, our final total is 970 + 76 + 51 + 26 + 1 = 1124, and the answer is 124.
