the problem is as follows, i have NO idea how to approach it or do it. any help is appreciated.

find all positive integers n such that n! ends with exactly 74 zeroes in decimal notation

Printable View

- Jun 15th 2009, 01:58 PMsilentbobquick question
the problem is as follows, i have NO idea how to approach it or do it. any help is appreciated.

find all positive integers n such that n! ends with exactly 74 zeroes in decimal notation - Jun 15th 2009, 02:36 PMSoroban
Hello, silentbob!

This takes a bit of brainwork . . .

Quote:

Find all positive integers such that ends with exactly 74 zeroes.

A final zero is created by a factor of 5 (and an even number).

The question becomes: how many factors of 5 are contained in ?

We know that every 5th number is a multiple of 5.

Also that every 25th number is a multiple of 5² = 25,

. . each of which contributes another 5.

And that every 125th number is a multiple of 5³ = 125,

. . each of which contributes yet another 5.

By trial-and error, we find that ends in 72 zeros,

. . but ends in 74 zeros.

. .

And the same is true for

. . But ends in 75 zeros.

Therefore: .