Question

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

• A. 5
• B. 7
• C. 4
• D. 6

Answer 1

The Correct Option is B

We need to find the largest positive integer \( n \) such that:

\[ (n!)! \mid (15000)! \]

This means the factorial of \( n! \) must divide \( 15000! \). Since factorials grow very quickly, we expect this to only hold for relatively small values of \( n \).

We try values of \( n \) such that \( (n!)! \leq 15000! \). In other words, we must ensure:

\[ n! \leq 15000 \]

• \( n = 5 \Rightarrow 5! = 120 \Rightarrow (120)! \leq (15000)! \) ✅
• \( n = 6 \Rightarrow 6! = 720 \Rightarrow (720)! \leq (15000)! \) ✅
• \( n = 7 \Rightarrow 7! = 5040 \Rightarrow (5040)! \leq (15000)! \) ✅
• \( n = 8 \Rightarrow 8! = 40320 \Rightarrow (40320)! > (15000)! \) ❌

The largest integer \( n \) such that \( (n!)! \) divides \( (15000)! \) is:

\[ \boxed{7} \]

Answer 2

Objective: 
To find the greatest possible value of \(n\) such that \(n! < 15000\)

Step 1: Compute factorial values
\(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)
\(8! = 8 \times 7! = 8 \times 5040 = 40320\)

Since \(7! = 5040 < 15000\) and \(8! = 40320 > 15000\),
the maximum possible value of \(n\) such that \(n! < 15000\) is:

Final Answer: \(n = 7\)
Correct Option: (B)