CAT Quantitative Aptitude Questions | CAT Number Systems - Factorial questions

The question is about Factorial, factors, primes. We need to find out the possible values of n if (n + 1) is a factor of n!. Dealing with factorials of a number is a vital component in CAT Number Theory in the CAT Exam. A range of CAT questions can be asked based on this simple concept in the CAT exam. Factorials are also useful in Permutation Combination questions in CAT. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz,head on here to take these questions in a test format, absolutely free.

Question 2: Given N is a positive integer less than 31, how many values can n take if (n + 1) is a factor of n!?

1. 18
2. 16
3. 12
4. 20

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factorial: A number that is not prime can be writen as a * b where a and b are natural numbers not equal to 1. Blinding insight using which we can solve this question.

The best starting point for this question is to do some trial and error.
3 is not a factor of 2!
4 is not a factor of 3!
5 is not a factor of 4!

6 is a factor of 5!
7 is not a factor of 6!
8 is a factor of 7!

The first thing we see is that n + 1 cannot be prime. If (n + 1) were prime, it cannot be a factor of n!.

So, we can eliminate all primes.

Now, let us think of all numbers where (n + 1) is not prime. In this instance, we should be able to write (n + 1) as a * b where a,b are not 1 and (n + 1). So, (a, b) will lie in the set {1, 2, 3........n} or, a * b will be a factor of (n + 1)!

So, for any composite (n + 1), (n + 1) will always be a factor of n! {Is there any exception?}

For any prime number (n + 1), (n + 1) will never be a factor of n!

The above rule works well even for all the examples we have seen, except when (n + 1) = 4. 4 = 2 * 2; So, 4 is not a factor of 3!. But this is the only exception.

Counting on from here, we can see that n can take values 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29. Essentially, all numbers where N + 1 is greater than 4 and is not prime will feature in this list. If N + 1 is not prime, we should be able to write it as a product of 2 numbers less than N + 1. This will feature in n!. This question is just a different way of asking one to count primes (and then account for the exception of 4)

n can take 18 different values.

The question is "how many values can n take if (n + 1) is a factor of n!?"

Hence the answer is "18"

Choice A is the correct answer.