The largest natural number n such that 3n 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[\dfrac{a}{p}\right] + \left[\dfrac{a}{{p}^{2}}\right] + \left[\dfrac{a}{{p}^{3}}\right] + \left[\dfrac{a}{{p}^{4}}\right] + . . .$</p><p>Since,$3$is a prime number, so we have</p><p>$= \left[\dfrac{66}{3}\right] + \left[\dfrac{66}{{3}^{2}}\right] + \left[\dfrac{66}{{3}^{3}}\right] + \left[\dfrac{66}{{3}^{4}}\right] + . . .$</p><p>$= 22 + 7 + 2 + 0$</p><p>$= 31$</p><p>So, maximum value of$n$is$31.