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The question is a question on factors/divisors of a number in the topic number theory.

If both 11² and 3³ are factors of the number a * 4³ * 6² * 13¹¹, then what is the smallest possible value of a?

A. 121
B. 3267
C. 363
D. 33
E. None of the above

The correct choice is (C) and the correct answer is 363.

11² is a factor of the given number.

The number does not have a power or multiple of 11 as its factor.

Hence, "a" should include 11²

3³ is a factor of the given number. 6² is a part of the number. 6² has 3² in it.

Therefore, if 3³ has to be a factor of the given number a * 4³ * 6² * 13¹¹, then we will need at least another 3.

Therefore, "a" should be at least 11² * 3 = 363 if the given number has to have 11² and 3³ as its factors.

The smallest value that "a" can consequently take is 363.

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