Results 1 to 4 of 4

Math Help - [SOLVED] How many 0's at the end of 100!

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    33

    [SOLVED] How many 0's at the end of 100!

    How many 0's are at the end of 100! ?
    Can someone check if this is correct? Thanks.

    My reasoning: numbers that produce 0's at the end are 2, 5 and any other integers 2^r < 100 , 5^s < 100
    2^1 = 2
    2^2 = 4
    2^3 = 8
    2^4 = 16
    2^5 = 32
    2^6 = 64
    5^1 = 5
    5^2 = 25
    r + s = 1 + 2 + 3 + 4 + 5 + 6 + 1 + 2 = 24
    There are 24 0's at the end of 100!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,864
    Thanks
    744
    Hello, VENI!

    Amazing . . . you did it all wrong and got the right answer!


    How many 0's are at the end of 100! ?
    It has nothing to do with the powers-of-2.


    Question: how many factors-of-5 are in 100-factorial?

    Reasoning: Each factor-of-5, paired with any even number, will produce a zero at the end.

    Since half the numbers are even, there are plenty of even numbers available.

    So how many factors-of-5 are there?


    Well, every fifth number is a multiple of 5.
    . . So there are: . \frac{100}{5} \,=\,20 of them.

    But some of them are multiples of 5^2=25, which provide an extra factor of 5.
    . . And there are: . \frac{100}{25} = 4 of them.


    Hence, there are: . 20+4\:=\:24 factors-of-5 in 100-factorial.

    Therefore, it ends in 24 zeros.

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2009
    Posts
    33
    Haha, what a coincidence.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Mar 2007
    Posts
    1,240

    Talking

    Quote Originally Posted by VENI View Post
    How many 0's are at the end of 100! ?
    Some online articles explain the general method, so you can find the number of zeroes for any factorial.

    Have fun!
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum