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Math Help - What is the greates positive integer...

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    What is the greates positive integer...

    What is the greatest positive integer n such that 3^n is a factor of 200! ?
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    Quote Originally Posted by ceasar_19134 View Post
    What is the greatest positive integer n such that 3^n is a factor of 200! ?
    What you actually want to know is how many times is 200! factorial divisible by 3.

    Now, there is a theorem from number theory that says, (by a mathemation I respect, Legendre)
    <br />
\boxed{n!=\prod_{p\leq n} p^{\sum _{k=1} ^{\infty} [n/p^k]}}

    I know, I know, one scary looking formula.
    But what it is actually saying is that to find that value you need to compute,
    \sum_{k=1}^{\infty} \left[ \frac{n}{p^k} \right]
    Where [ \, ] is the greatest integer function.
    Now, here n!=200! thus, n=200. And the prime is p=3.
    Thus,
    \sum_{k=1}^{\infty} \left[ \frac{200}{3^k} \right]
    This sum is actually finite!
    Because when the exponent gets large enough the greatest integer will be zero. Thus you will just be summing up zeros.
    Thus,
    [200/3]+[200/9]+[200/27]+[200/81]+[200/243]+...
    The way you evaluate is by dividing and dropping the decimal part.
    66+22+7+2+0+0+0+...
    Thus it is,
    97
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