The largest natural number n such that 3n divides 66! is _______ . [2023]

Question

The largest natural number n such that $3^{n}$ divides 66! is _______ . [2023]

Smart Achievers · Accepted Answer

(31)

The largest prime $p$ in $n!$ is

$E_{p} (n!) = [\frac{n}{p}] + [\frac{n}{p^{2}}] + [\frac{n}{p^{3}}] + \dots$

Here, $p$ = 3 is a prime number and $n$ = 66

$∴$ $E_{3} (66!) = [\frac{66}{3}] + [\frac{66}{9}] + [\frac{66}{27}] + [\frac{66}{81}] + \dots$

$= 22 + 7 + 2 + 0 + \dots = 31$ $∴$ $66! = 3^{31} \dots$

$∴ The maximum exponent of 3 is 31.$

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