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CAT 2022 Slot 2 — QA Question 9

FactorialsEasy

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

Answer & solution

  • A

    6

  • B

    5

  • 7

  • D

    4

Solution

Easy

For (15000)!(15000)! to be divisible by (n!)!(n!)!, it is enough — and necessary here — that the inner factorial argument fits: n!≤15000n!\le 15000, because m!m! divides (15000)!(15000)! whenever m≤15000m\le 15000. So find the largest nn with n!≤15000n!\le15000.

1

Key reduction. (n!)!=(m)!(n!)!=(m)! where m=n!m=n!. And m!∣(15000)!m!\mid(15000)! iff m≤15000m\le15000. So we need

n!≤15000.n!\le 15000.
2

Check factorials:

6!=7207!=5040 ≤15000 ✓8!=40320 >15000 ×\begin{aligned} &6!=720\\ &7!=5040\ \le 15000\ \checkmark\\ &8!=40320\ >15000\ \times \end{aligned}

Largest valid n=7n=\mathbf{7} — option (c).

CAT 2022 Slot 2 QA Q9: For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is: — Solution | TheCATExam