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.

22 and 4

32 and 2

24 and 3

25 and 4

Correct Answer

Choice C. 24 and 3

Detailed Solution

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.

Correct Answer: 24 and 3

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