CAT Quantitative Aptitude Questions | CAT Number Systems - Factorial questions

The question is about different bases and factorials. Number of zeroes at the end of 'n' is given for different bases and we need to find out the possible values of 'n' . Dealing with factorials of a number is a vital component in CAT Number Systems. 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 1: A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions.

1. 22 and 4
2. 32 and 2
3. 24 and 3
4. 25 and 4

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factorial: Number Theory lets us combine base 6, base 8 math with the idea of factorials. Why would we not want to do that?

Base 6 representation ends with 10 zeroes, or the number is a multiple of 6¹⁰. If n! has to be a multiple of 6¹⁰, it has to be a multiple of 3¹⁰. The smallest factorial that is a multiple of 3¹⁰ is 24!. So, when n = 24, 25 or 26, n! will be a multiple of 6¹⁰ (but not 6¹¹).

Similarly, for the second part, we need to find n! such that it is a multiple of 2²¹, but not 2²⁴. When n = 24, n! is a multiple of 2²². S0, when n = 24, 25, 26, 27, n! will be a multiple of 2²¹ but not 2²⁴.

The smallest n that satisfies the above conditions is 24. n = 24, 25 or 26 will satisfy the above conditions.

The question is "Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions."

Hence the answer is "24 and 3"

Choice C is the correct answer.