CAT Quantitative Aptitude Questions | CAT Number Systems - Factorial questions

The question is about basic factorial. We need to find out the possible values of 'n', when n! is a multiple of a^x but not a^y. Dealing with factorials of a number is a vital component in CAT Number Theory. 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 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

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factorial: Finding the highest power of 7 that divides n! is easy. Find the smallest n such that n! is a multiple of 7³² might not be easy. Think about that...

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 [\\frac{n}{7}\\)] + [\\frac{n}{49}\\)] + [\\frac{n}{343}\\)] + ...........
where [x] is the greatest integer less than or equal to x.

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.

The question is "How many values can natural number n take, if n! is a multiple of 7⁶ but not 7⁹?"

Hence the answer is "14 values"

Choice C is the correct answer.