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)
I know, I know, one scary looking formula.
But what it is actually saying is that to find that value you need to compute,
Where is the greatest integer function.
Now, here thus, . And the prime is .
Thus,
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,
The way you evaluate is by dividing and dropping the decimal part.
Thus it is,