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

The correct answer is B: 7
To find the largest possible value of n such that \((15,000)!\) is divisible by \((n!)!\),we can follow these steps: 
1. Let's assume \(n! = k\),where k is a positive integer. 
2. We are given that \((15,000)!\) is divisible by \((n!)!\).This implies that \(k!\) divides \((15,000)!\). 
3. Therefore,we need to find the largest value of \(k (or\space n!)\) such that \(k!\) divides \((15,000)!\). 
4. Let's calculate some factorials to determine the value of k: 
- \(5! = 120\)
- \(6!=720\)
- \(7!=5,040\) 
- \(8!=40,320\)
5. As we can see, when \(k (or\space n!)\) is 7,we have \(k! = 5,040\), which divides evenly into \(15,000!\).This means that \((7!)! = 5,040!\space divides\space(15,000)!\). 
6. Now,let's check for the next value of k,which is 8: 
-\(8!=40,320\)
- However,40,320 is not a factor of \(15,000!\) (since it's larger than 15,000). 
7. Therefore,the largest possible value of n (or k) is 7,as it's the highest value for which \(k!\) divides \(15,000!\). 
Hence, the answer is indeed n = 7.

Answer 2

In order to determine the greatest possible value of \(n\), we must find the value of n at which \(n!\) is less than \(15000\).
\(7! = 7 \times 6\times5\times4\times3\times2 \times 1\)
\(7!= 5040\)
and,
\(8! =8  \times 7 \times 6\times5\times4\times3\times2 \times 1\)
\(8!=40320\)

\(⇒\) This indicates that \(15000!\) is not divisible by \(40320!\)
Hence, the highest value \(n =7\).

So, the correct option is (B): \(7\)