JEE MAIN - Mathematics (2023 - 8th April Morning Shift - No. 26)
The largest natural number $$n$$ such that $$3^{n}$$ divides $$66 !$$ is ___________.
Answer
31
Explanation
We have,
$$ \begin{aligned} & {\left[\frac{66}{3}\right]=22} \\\\ & {\left[\frac{66}{3^2}\right]=7} \\\\ & {\left[\frac{66}{3^3}\right]=2} \end{aligned} $$
Highest powers of 3 is greater than 66. So, their g.i.f. is always 0.
$\therefore$ Required natural number $=22+7+2=31$
$$ \begin{aligned} & {\left[\frac{66}{3}\right]=22} \\\\ & {\left[\frac{66}{3^2}\right]=7} \\\\ & {\left[\frac{66}{3^3}\right]=2} \end{aligned} $$
Highest powers of 3 is greater than 66. So, their g.i.f. is always 0.
$\therefore$ Required natural number $=22+7+2=31$
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