CAT Practice : Number System: Factorial

You are here: Home  CAT Questionbank   CAT Quant  Number System: Factorial  Question 4
Finding the highest power of 3 that divides n! is easy. Find the smallest n such that n! is a multiple of 3^k might not be easy. Think about that.

Factorials - basic

    Q.4: How many values can natural number n take, if n! is a multiple of 220 but not 320?
    1. 11
    2. 21
    3. 16
    4. 5

 

  • Correct Answer
    Choice B. 21

Detailed Solution

The highest power of 2 that will divide n! = ........ and so on. So, let us try to find the smallest n such that n! is a multiple of 220.

If n = 10, the highest power of 2 that will divide n! = = 5 + 2 + 1 = 8

If n = 20, the highest power of 2 that will divide n! = = 10 + 5 + 2 + 1 = 18

If n = 24, the highest power of 2 that will divide n! = = 12 + 6 + 3 + 1 = 22

{Here we can also see that each successive number is just the quotient of dividing the previous number by 2. As in, = 6, = 3, = 1. This is a further short-cut one can use.}

So the lowest number of n such that n! is a multiple of 220 is 24.

Now, moving on to finding n! that is a multiple of 3. The highest power of 3 that will divide n! = and so on.

When n = 20, the highest power of 3 that can divide 20! = = 6 + 2 = 8
When n = 35, the highest power of 3 that can divide 35! = = 11 + 3 + 1 = 15
When n = 45, the highest power of 3 that can divide 45! = = 15 + 5 +1 = 21

The lowest number n such that n! is a multiple of 320 is 45.

When n takes values from 24 to 44, n! will be a multiple of 220 and not 320. n can take 21 values totally.

Correct Answer: 21 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.