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?

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
add two 625s and
7500! ends in 1874 zeroes
add three 125s and
7875! ends in 1967 zeroes
add four 25s and
7975! ends in 1991 zeroes
add one 5 and
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