IMO the Answer is B. The way I solved it is --

since x is divisible by 6 so x = 6M

hence sqrt288kx= 24sqrt3kM

now except for option B all other options can be derived by putting some integer value of M . Only for option B M needs to be 1/3. but the condition already states that M is an integer

I also got B.

First, it's critical to know that 24^2 = 576, and also that 288 x 2 = 576. Remarking those two points is critical to solving this problem.

Next, just ignore k, it doesn't affect anything.

next, see which of the answers can be factored into the 288*X*K format:

A) 24k*[root 3]
= square root [576 * k^2 * 3]
= square root [288 * 2 * 3 * k^2]

2*3 = 6, so there is your X. X = 6 in this case, which is obviously divisible by 6

B) 24*[root k]
= square root [576 * k]
= square root [288 * 2 * k]

2 is our X value here, and it is clearly not divisible by 6, so this value cannot be put into the format in such a way that the problem's criteria are met.

if you're second doubting yourself, or me, you could keep going I guess:

C) 24*[root 3k]
= square root [576 * 3 * k]
= square root [288 * 2 * 3 * k]

here, again, 2 * 3 = 6 which is our X value, which is divisible by 6.

D) 24*[root 6k]
= square root [576 * 6 * k]
= square root [288 * 2 * 6 * k]

here, 2 * 6 = 12 which is our X value, which is divisible by 6

E) 72*[root k]
= square root [5184 * k]
= square root [288 * 18 * k]

here, 18 is our X value, which is divisible by 6.

I hope this was clear, and more importantly, right.