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

Question

UPSC Quizroom · Accepted Answer

Step 1: Translate the problem

We want (n!)! to divide 15,000!.

For k! to divide m!, we must have k ≤ m.

Here, (n!)! is the factorial of n!. So we need:

(n!)! ≤ 15,000!

We are looking for the largest n such that this inequality holds.

Step 2: Approximate using factorial growth

Factorials grow rapidly. Recall:

k! grows very fast with k, so (n!)! ≤ 15,000! implies n! ≤ 15,000

Step 3: Find largest n with n! ≤ 15,000

n = 5 → 5! = 120 ≤ 15,000 ✅

n = 6 → 6! = 720 ≤ 15,000 ✅

n = 7 → 7! = 5040 ≤ 15,000 ✅

n = 8 → 8! = 40,320 > 15,000 ❌

So the largest n satisfying n! ≤ 15,000 is n = 7.

Answer:

7

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