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Quantitative Aptitude Question on Divisibility and Factors

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

A

5

B

7

C

4

D

6

Answer

7

Explanation

Solution

The correct answer is B: 7
To find the largest possible value of n such that (15,000)!(15,000)! is divisible by (n!)!(n!)!,we can follow these steps:
1. Let's assume n!=kn! = k,where k is a positive integer.
2. We are given that (15,000)!(15,000)! is divisible by (n!)!(n!)!.This implies that k!k! divides (15,000)!(15,000)!.
3. Therefore,we need to find the largest value of k(or  n!)k (or\space n!) such that k!k! divides (15,000)!(15,000)!.
4. Let's calculate some factorials to determine the value of k:

  • 5!=1205! = 120
  • 6!=7206!=720
  • 7!=5,0407!=5,040
  • 8!=40,3208!=40,320
    5. As we can see, when k(or  n!)k (or\space n!) is 7,we have k!=5,040k! = 5,040, which divides evenly into 15,000!15,000!.This means that (7!)!=5,040!  divides  (15,000)!(7!)! = 5,040!\space divides\space(15,000)!.
    6. Now,let's check for the next value of k,which is 8:
    -8!=40,3208!=40,320
    - However,40,320 is not a factor of 15,000!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!k! divides 15,000!15,000!.
    Hence, the answer is indeed n = 7.