What is the greates positive integer...

**ceasar_19134** · December 13th 2006, 04:42 PM

What is the greatest positive integer n such that 3^n is a factor of 200! ?

**ThePerfectHacker** · December 13th 2006, 04:58 PM

Originally Posted by ceasar_19134

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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