Number System Question 3 - CAT 2022
For some natural number n, assume that 15000! is divisible by (n!)!. The largest possible value of n is:
Step-by-Step Explanation
For one factorial to divide another factorial perfectly, the number inside must be smaller or equal. For example, 10! easily divides 15!, but 20! cannot divide 15!.
In this question, we want 15000! to be divisible by (n!)!. This means the number n! must be less than or equal to 15000:
n! ≤ 15000
Let's write down the standard factorials we know to see where we cross the line:
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 6! = 6 × 120 = 720
- 7! = 7 × 720 = 5040
- 8! = 8 × 5040 = 40320
Since 40320 is way past 15000, n cannot be 8. The largest safe value n can take is 7, because 7! = 5040, which is comfortably below 15000.
Correct Option: B
💡 FundaMakers Tip:
The double factorial looks intimidating, but you can ignore the outer layer. The question is simply asking: "What is the biggest number whose single factorial stays under 15,000?" Listing your basic factorials up to 8 makes the choice obvious.