The largest natural number n such that 3 n divides 66 ! is _ _ _ _ _ _ _

We know that,

Maximum value of $n$ for $p^n$ where $p$ is prime divides the number $a!$ is given by,

 $\left[\frac{a}{p}\right]+\left[\frac{a}{p^2}\right]+\left[\frac{a}{p^3}\right]+\left[\frac{a}{p^4}\right]+...$

Since, $3$ is a prime number, so we have

$=\left[\frac{66}{3}\right]+\left[\frac{66}{3^2}\right]+\left[\frac{66}{3^3}\right]+\left[\frac{66}{3^4}\right]+...$

$=22+7+2+0$

$=31$

So, maximum value of $n$ is $31$.