Question

Maximum value n such that (66)! is divisible by $3^n$

Answer 1

To find the maximum value of n such that (66)! is divisible by 3n, we need to count the number of factors of 3 in (66)!, since each factor of 3 contributes to the divisibility by 3.
To count the number of factors of 3 in (66)!, we can use the formula:
$[\frac{66}{3}]+[\frac{66}{9}]+[\frac{66}{27}]+[\frac{66}{81}]$
= 22 + 7 + 2 + 0 = 31,
where ⌊x⌋ denotes the greatest integer less than or equal to x.
This means that (66)! is divisible by 331. Therefore, the maximum value of n such that (66)! is divisible by $3^n$ is n = 31.
So, the correct answer is 31