The largest natural number n such that 3^{n} | vedclass

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Question

(B) To find the largest power of a prime $p$ that divides $n!$,we use Legendre's formula: $E_p(n!) = \sum_{k=1}^{\infty} \left[ \frac{n}{p^k} \right]$

Vedclass · Accepted Answer

$31$

Vedclass · Answer

(B) To find the largest power of a prime $p$ that divides $n!$,we use Legendre's formula: $E_p(n!) = \sum_{k=1}^{\infty} \left[ \frac{n}{p^k} \right]$.Here,$n = 66$ and $p = 3$.$E_3(66!) = \left[ \frac{66}{3} \right] + \left[ \frac{66}{3^2} \right] + \left[ \frac{66}{3^3} \right]$$E_3(66!) = \left[ \frac{66}{3} \right] + \left[ \frac{66}{9} \right] + \left[ \frac{66}{27} \right]$$E_3(66!) = 22 + 7 + 2 = 31$.Thus,the largest natural number $n$ is $31$.

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

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