CAT 2022 Slot 2 QA Question 1: For some natural number $n$, assume that $(15,000)!$ is divisible by $(n!)!$. The largest possible value of $n$ is

Question

For some natural number $n$, assume that $(15,000)!$ is divisible by $(n!)!$. The largest possible value of $n$ is

Accepted Answer

For any two positive integers $k$ and $m$, $k!$ divides $m!$ whenever $k \leq m$.

Here, $(n!)!$ will divide $(15,000)!$ if $n! \leq 15,000$.

$7! = 5040 \leq 15,000$
$8! = 40,320 > 15,000$

Hence, the largest possible value of $n$ is 7.

Answer: (3) 7

For some natural number nnn, assume that (15,000)!(15,000)!(15,000)! is divisible by (n!)!(n!)!(n!)!. The largest possible value of nnn is