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 Curious Consecutive Cyphers II (Posted on 2009-02-23)
A positive integer T is defined as a factorial tail if there exists a positive integer P such that the decimal expansion of P! ends with precisely T consecutive zeroes, and (T+1)th digit from the right in P! is nonzero.

How many positive integers less than 1992 are not factorial tails?

 No Solution Yet Submitted by K Sengupta No Rating

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 Complete Analytic Solution | Comment 6 of 7 |
With factorial each multiple of 5 adds a zero
each multiple of 5^n adds n zeroes and so skips (n-1)

The first thing we need is the largest factorial with 1992 zeroes or fewer.

5! ends in 1 zero
25! ends in 5+1 = 6 zeroes
125! ends in 25+5+1 = 31 zeroes
625! ends in 156 zeroes
3125! ends in 781 zeroes [the next power has too many]

(2*3125)! = 6250! ends in 2*781=1562 zeroes
7500! ends in 1874 zeroes
7875! ends in 1967 zeroes
7975! ends in 1991 zeroes
7980! ends in 1992 zeroes

[7980/25] = 319 skipped tails where brackets indicate the greatest integer
[7980/125] = 63 extra skipped tails
[7980/625] = 12 extras
[7980/3125] = 2 extras

For a total of 319+63+12+2 = 396 numbers that are not factorial tails.

 Posted by Jer on 2009-02-24 14:23:54

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