Question

What is the maximum value of n for which 352! is divisible by 352^n​

Answer 1

First we need to find prime factors of 352.

$\displaystyle\sf{\longrightarrow 352=2^5\times 11}$

The prime factors are 2 and 11. We need to consider the largest among them, i.e., 11.

We see $\displaystyle\sf {11^2<352<11^3.}$

So the maximum value of $\displaystyle\sf {n}$ is given by,

$\displaystyle\sf{\longrightarrow n=\lfloor\dfrac {352}{11}\rfloor+\lfloor\dfrac {352}{11^2}\rfloor}$

$\displaystyle\sf{\longrightarrow n=\lfloor\dfrac {352}{11}\rfloor+\lfloor\dfrac {352}{121}\rfloor}$

$\displaystyle\sf{\longrightarrow n=32+2}$

$\displaystyle\sf {\longrightarrow\underline {\underline {n=34}}}$

Hence 34 is the answer.