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Problem 383


Problem 383


Divisibility comparison between factorials

Let f5(n) be the largest integer x for which 5x divides n.
For example, f5(625000) = 7.

Let T5(n) be the number of integers i which satisfy f5((2·i-1)!) < 2·f5(i!) and 1 ≤ i ≤ n.
It can be verified that T5(103) = 68 and T5(109) = 2408210.

Find T5(1018).


阶乘之间的整除性比较

记f5(n)为使得5x整除n的最大整数x。
例如,f5(625000) = 7。

记T5(n)为满足f5((2·i-1)!) < 2·f5(i!)且1 ≤ i ≤ n的整数i的数目。
可以验证,T5(103) = 68以及T5(109) = 2408210。

求T5(1018)。