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

Question

The largest natural number $n$ such that $3^{n}$ divides $66 !$ is \_\_\_\_\_\_\_\_\_\_\_.

Answer: 31

Accepted Answer

We have,

$\begin{gathered} [\frac{66}{3}]=22 \ [\frac{66}{{3}^{2}}]=7 \ [\frac{66}{{3}^{3}}]=2 \end{gathered}$

Highest powers of 3 is greater than 66. So, their g.i.f. is always 0.

$	herefore $ Required natural number $=22+7+2=31$

Solutions · 1