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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