# CAT Practice : Number System: Factorial

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Number Theory lets us combine base 6, base 8 math with the idea of factorials. Why would we not want to do that?

## Factorial Base 6 base 8 math

Q.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

Choice C. 24 and 3

## Detailed Solution

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.

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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.