So for this problem, my approach was to start with the largest answer and to see if it (in this case 12) cubed would be divisible by 24 using the primes approach.

If you can interpret my scratch work, I found that 12 cubed is divisible by 24 and thus marked it as the correct answer. However, the book indicated that 6 is the correct answer. Can someone tell me what I’m missing? Thanks in advance.

Since it is mentioned that n^3 is divisible by 24, n^3 = 24* something (let’s say something is z). Now, n^3 = 24* z. If we take the square root of the same, the number will be n= cube-root of 24* z.
Now cube-root of 24 is 2* cube-root(3) and for n it will be 2* cube-root(3*z). Also, If you noticed, it is mentioned that n is an integer so for the whole n to be an integer, you would need the number to be at least equal to 27 (i.e z=9). That would make the number n = 2 * 3. Factors of 6 will help you with the highest factor which is 6.