The largest natural number $$n$$ such that $$3^n$$ divides 66! is ______.

To find the largest exponent $$n$$ such that $$3^n$$ (reading $$3n$$ as $$3^n$$ based on the context of such factorial problems) divides $$66!$$, we use Legendre's Formula.

Legendre's Formula

The exponent of a prime $$p$$ in $$m!$$ is given by:

$$E_p(m!) = \left\lfloor \frac{m}{p} \right\rfloor + \left\lfloor \frac{m}{p^2} \right\rfloor + \left\lfloor \frac{m}{p^3} \right\rfloor + \dots$$

Calculation for $$p=3$$ and $$m=66$$

1. $$\lfloor 66/3 \rfloor = 22$$
2. $$\lfloor 66/9 \rfloor = 7$$
3. $$\lfloor 66/27 \rfloor = 2$$
4. $$\lfloor 66/81 \rfloor = 0$$ (Stop here)

Sum: $$22 + 7 + 2 = 31$$