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

Question

Let \(n\) be the least positive integer such that \(168\) is a factor of \(1134^n\) . If \(m\) is the least positive integer such that \(1134^n\) is a factor of \(168^m\) , then \(m+ n \) equals

Answer 1

Determine the maximum positive integer \( n \) satisfying the divisibility condition:

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

This implies that \( (n!)! \) must be a divisor of \( 15000! \). Given the rapid growth of factorials, \( n \) is expected to be small.

The condition \( (n!)! \leq 15000! \) simplifies to:

\[ n! \leq 15000 \]

For \( n = 5 \), \( 5! = 120 \). Since \( 120 \leq 15000 \), \( (120)! \leq (15000)! \). ✅
For \( n = 6 \), \( 6! = 720 \). Since \( 720 \leq 15000 \), \( (720)! \leq (15000)! \). ✅
For \( n = 7 \), \( 7! = 5040 \). Since \( 5040 \leq 15000 \), \( (5040)! \leq (15000)! \). ✅
For \( n = 8 \), \( 8! = 40320 \). Since \( 40320>15000 \), \( (40320)!>(15000)! \). ❌

The largest integer \( n \) for which \( (n!)! \) divides \( (15000)! \) is:

\[ \boxed{7} \]

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

The Correct Option is B

Solution and Explanation

Top Questions on Divisibility and Factors

Questions Asked in CAT exam