-------ASIDE---------------------------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving factors, we can say:

If k is a factor of N, then k is "hiding" within the prime factorization of N

Consider these examples:
3 is a factor of 24, because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a factor of 70 because 70 = (2)(5)(7)
And 8 is a factor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a factor of 630 because 630 = (2)(3)(3)(5)(7)
-----BACK TO THE QUESTION!---------------------

So, the original question can be rephrased to ask "What's the smallest value of n so that 288 is hiding in the prime factorization of 6^n?
Let's find out..

288 = (2)(2)(2)(2)(2)(3)(3)
So, in order for 288 to be a factor of 6^n, the prime factorization of 6^n must contain at least five 2's and two 3's

So, for example, 6⁴ = (6)(6)(6)(6) = (2)(3)(2)(3)(2)(3)(2)(3).
As you can see, 6⁴ contains at least two 3's, BUT it doesn't contain the five 2's we need.
So, 288 is not a factor of 6⁴

On the other hand, 6⁵ = (6)(6)(6)(6)(6) = (2)(3)(2)(3)(2)(3)(2)(3)(2)(3) = (2)(2)(2)(2)(2)(3)(3)(3)(3)(3), which means 288 is definitely a factor of 6⁵

Answer: D