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.

- 22 and 4
- 32 and 2
- 24 and 3
- 25 and 4

24 and 3

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Base 6 representation ends with 10 zeroes, or the number is a multiple of 6^{10}. If n! has to be a multiple of 6^{10}, it has to be a multiple of 3^{10}. The smallest factorial that is a multiple of 3^{10} is 24!. So, when n = 24, 25 or 26, n! will be a multiple of 6^{10} (but not 6^{11}).

Similarly, for the second part, we need to find n! such that it is a multiple of 2^{21}, but not 2^{24}. When n = 24, n! is a multiple of 2^{22}. S0, when n = 24, 25, 26, 27, n! will be a multiple of 2^{21} but not 2^{24}.

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."**

Choice C is the correct answer.

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