Question

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

Accepted Answer

B. We want $A!$ to be divisible by $B!$ where $A = 15,000$ and $B = n!$. For $A!$ to be divisible by $B!$, we need $B \le A$, so $n! \le 15,000$. Let's calculate $n!$ for small values of $n$: $1! = 1$ $2! = 2$ $3! = 6$ $4! = 24$ $5! = 120$ $6! = 720$ $7! = 5040$ $8! = 40320 > 15,000$. Since $7! = 5040 \le 15,000$, $(5040)!$ divides $(15,000)!$. Thus, the maximum possible natural number $n$ is 7.

Divisibility of Factorials

Explanation

Practise this under exam conditions