Problem Solving

If k and x are positive integers and x is divisible by 6, which of the following CANNOT be the value of sqrt (288kx) ?

A) 24k sqrt(3)
B) 24 sqrt(k)
C) 24 sqrt(3k)
D) 24 sqrt(6k)
E) 72 sqrt(k)

What would be your best approach to solve this problem quickly and accurately ?
Thanks.

Please feel free to state any secondary strategies you would use to answer this incase you were running out of time, or came stuck on this problem.

Thanks in advance.

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.