CAT Practice : Number System: Factorial

You are here: Home  CAT Questionbank   CAT Quant  Number System: Factorial  Question 3
Finding the highest power of 7 that divides n! is easy. Find the smallest n such that n! is a multiple of 7^32 might not be easy. Think about that...

Factorials - basic

    Q.3: How many values can natural number n take, if n! is a multiple of 76 but not 79?
    1. 7
    2. 21
    3. 14
    4. 12


  • Correct Answer
    Choice C. 14

Detailed Solution

The smallest factorial that will be a multiple of 7 is 7!
14! will be a multiple of 72
Extending this logic, 42! will be a multiple of 76

However, 49! will be a multiple of 78 as 49 (7 * 7) will contribute two 7s to the factorial. (This is a standard question whenever factorials are discussed). Extending beyond this, 56! will be a multiple of 79.

In general for any natural number n,
n! will be a multiple of + ...........
where [x] is the greatest integer less than or equal to x. A more detailed discussion of this is available on this link

So, we see than 42! is a multiple of 76. We also see that 56! is the smallest factorial that is a multiple of 79. So, n can take values {42, 43, 44, 45........55}

There are 14 values that n can take.

Correct Answer: 14 values

Our Online Course, Now on Google Playstore!

2IIM's App

Fully Functional Course on Mobile

All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.

Cache Content for Offline Viewing

Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!

Get it on Google Play

More questions from Number System - Factorial

  1. Factorial Base 6, Base 8 math
  2. Factorial, factors
  3. Factorials - Basic
  4. Factorials - Basic
  5. Factorials - Trailing zeroes
  6. Factorials - Basic
  7. Factorials - Trailing zeroes
  8. Factorials - Base 8
  9. Factorials and Factors
This idea is so good that it comes with an exclamation mark. N! holds marvels that you might not have noticed before. Enter here to see those.