The largest such that divides 50! is : [2025]

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Solution

Q.
The largest $n \in N$ such that $3^{n}$ divides $50!$ is : [2025]

1 23

2 22

3 21

4 20

Ans.
(2)

The number of times a prime p divides $n!$ is $\sum_{k = 1}^{\infty} [\frac{n}{p^{k}}]$

Required largest value of

$n = [\frac{50}{3}] + [\frac{50}{3^{2}}] + [\frac{50}{3^{3}}]$

= 16 + 5 + 1 = 22

Hence, the maximum value of $n$ is 22.

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