# of Numbers in Factorials

 

To find the maximum # of Numbers in a Factorial, best to look at an example.

$$\frac{100!}{3^x}=\textrm{integer}$$

This question is saying that $100!$ is divisible by $3^x$, where $x$ is an integer. So what would be the maximum value of $x$? In other words, how many powers of $3$ are there in $100!$?

Here's the Trick

• Step 1: How many multiples of $3^1$ are there from $1$ to $100$? $33$
• Step 2: How many multiples of $3^2$ ($9$) are there from $1$ to $100$? $11$
• Step 3: How many multiples of $3^3$ ($27$) are there from $1$ to $100$? $3$
• Step 4: How many multiples of $3^4$ ($81$) are there from $1$ to $100$? $1$
• Step 5: How many multiples of $3^5$ ($243$) are there from $1$ to $100$? $0$
  • Once we hit $0$, we stop.

Simply add those numbers together to find the number of powers of $3$ in $100!$:

$$33 + 11 + 3 + 1 = 48$$

WARNING: The trick below only works if you're dividing by a prime number power. See the next entry/video for why.