Question

The largest natural number \(n\) such that \(3^n\) divides \(66!\) is:

Hint:

To find the highest power of a prime dividing \(n!\), use successive divisions by powers of the prime.

Answer 1

Correct Answer: 31

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

Answer 2

Step 1: Use Legendre’s formula.
The largest power of a prime \(p\) dividing \(n!\) is given by: \[ \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots \] Step 2: Apply for \(p = 3\) and \(n = 66\).
\[ \left\lfloor \frac{66}{3} \right\rfloor + \left\lfloor \frac{66}{9} \right\rfloor + \left\lfloor \frac{66}{27} \right\rfloor = 22 + 7 + 2 = 31. \] Final Answer: The largest \(n\) is 31.